Create the numbers 1 - 30 using the digits 2, 0, 1, 9 in this particular order!












8














Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.



The rules haven't changed:




  • Use all four digits exactly once in the order 2-0-1-9.

  • Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.

  • Parentheses and grouping (e.g. "19") are also allowed.

  • Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.

  • The modulus operator $(%, mod)$ is not allowed.

  • Rounding (e.g. 201/9=22) is not allowed.


I'm curious to see your creative solutions!



May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!



Happy New Year and greetings from Germany!
André






formation-of-numbers number-theory







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  • Will you be giving a green check to someone?
    – flashstorm
    7 hours ago










  • Of course I'll do.
    – André
    3 hours ago
















8














Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.



The rules haven't changed:




  • Use all four digits exactly once in the order 2-0-1-9.

  • Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.

  • Parentheses and grouping (e.g. "19") are also allowed.

  • Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.

  • The modulus operator $(%, mod)$ is not allowed.

  • Rounding (e.g. 201/9=22) is not allowed.


I'm curious to see your creative solutions!



May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!



Happy New Year and greetings from Germany!
André






formation-of-numbers number-theory







share|improve this question






















  • Will you be giving a green check to someone?
    – flashstorm
    7 hours ago










  • Of course I'll do.
    – André
    3 hours ago














8












8








8


1





Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.



The rules haven't changed:




  • Use all four digits exactly once in the order 2-0-1-9.

  • Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.

  • Parentheses and grouping (e.g. "19") are also allowed.

  • Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.

  • The modulus operator $(%, mod)$ is not allowed.

  • Rounding (e.g. 201/9=22) is not allowed.


I'm curious to see your creative solutions!



May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!



Happy New Year and greetings from Germany!
André






formation-of-numbers number-theory







share|improve this question













Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.



The rules haven't changed:




  • Use all four digits exactly once in the order 2-0-1-9.

  • Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.

  • Parentheses and grouping (e.g. "19") are also allowed.

  • Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.

  • The modulus operator $(%, mod)$ is not allowed.

  • Rounding (e.g. 201/9=22) is not allowed.


I'm curious to see your creative solutions!



May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!



Happy New Year and greetings from Germany!
André






formation-of-numbers number-theory




formation-of-numbers number-theory






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share|improve this question




share|improve this question










asked 11 hours ago









André

1,178716




1,178716












  • Will you be giving a green check to someone?
    – flashstorm
    7 hours ago










  • Of course I'll do.
    – André
    3 hours ago


















  • Will you be giving a green check to someone?
    – flashstorm
    7 hours ago










  • Of course I'll do.
    – André
    3 hours ago
















Will you be giving a green check to someone?
– flashstorm
7 hours ago




Will you be giving a green check to someone?
– flashstorm
7 hours ago












Of course I'll do.
– André
3 hours ago




Of course I'll do.
– André
3 hours ago










4 Answers
4






active

oldest

votes


















3














1




1 = 2^(0*19)




2




2 = 2 + (0*19)




3




3 = 2 + 0!^19




4




4 = 2 ^ (0! + 1 ^ 9)




5




-((2 + 0! + 1) - 9)




6




-((2 + 0 + 1) - 9))




7




-((2 + 0*1 - 9))




8




-((2 - 01) - 9)




9




2*0*1 + 9




10




2*0 + 1 + 9




11




2 + 0*1 + 9




12




2 + 0 + 1 + 9




13




2 + 0! + 1 + 9




14




(2 + 0!)! - 1 + 9




15




(2 + 0 + 1)! + 9




16




(2 + 0!)! + 1 + 9




17




20 - (1 * sqrt(9))




18




(2 + (0 * 1)) * 9




19




20 - 1^9




20




20 * 1^9




21




20 + 1^9




22




2 + 0! + 19




23




20 + 1 * sqrt(9)




24




20 + 1 + sqrt(9)




25




2 || (0! + 1 + sqrt (9))




explained:




Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.




26




2 || ((0! + 1) * sqrt(9))




27




(2 + 0 + 1) * 9




28




((2 + 0!) || 1) - sqrt(9)




29




(2 + (0 * 1)) || 9




30




20 + 1 + 9







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  • edited for clarity
    – flashstorm
    10 hours ago










  • @flashstorm Um... $3ne 2+0^{19}$
    – Frpzzd
    10 hours ago






  • 1




    was missing an !
    – flashstorm
    10 hours ago










  • Ding! Fries are done :)
    – flashstorm
    10 hours ago



















6















$$1=20-19$$
$$2=2+0cdot 19=20div(1+9)$$
$$3=2cdot 0cdot 1+sqrt{9}=-(2+0+1)!+9$$
$$4=2^{0-1+sqrt{9}}$$
$$5=20div (1+sqrt{9})$$
$$6=(2cdot 0cdot 1+sqrt{9})!$$
$$7=-2-0cdot 1+9$$
$$8=2^{0cdot 1+sqrt{9}}$$
$$9=2cdot 0cdot 1+9$$
$$10=2cdot 0+1+9=20-1-9$$
$$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
$$12=2+0+1+9=20+1-9$$
$$13=2+0!+1+9=2^{0!+1}+9$$
$$14=(2+0!)!-1+9$$
$$15=(2+0+1)!+9$$
$$16=2^{0+1+sqrt{9}}$$
$$17=20-sqrt{1cdot 9}$$
$$18=20+1-sqrt{9}$$
$$19=2cdot 0+19$$
$$20=2^0+19$$
$$21=20+1^9$$
$$22=2cdot (0!+1+9)$$
$$23=20+sqrt{1cdot 9}$$
$$24=2^{0-1+sqrt{9}}!=20+1+sqrt{9}$$
$$25=(2+0!)!+19$$
$$26=2+0+(1+sqrt{9})!$$
$$27=(2+0+1)^{sqrt{9}}$$
$$28=20-1+9$$
$$29=20+1cdot 9=20cdot 1+9$$
$$30=20+1+9$$




DONE!






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  • All finished now! :D
    – Frpzzd
    10 hours ago










  • Great :) But Spoiler-Tags would be nice ;)
    – André
    8 hours ago












  • @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
    – Frpzzd
    8 hours ago






  • 1




    @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
    – Frpzzd
    8 hours ago










  • Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
    – user1207177
    8 hours ago





















2














1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.



8:




$8 = sqrt{(2^{0!+1}!!)!! / (sqrt9)!}$




16:




$16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$




17:




$17 = (2 + 0! + 1)!! + 9$




18:




$18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^{0+1} cdot 9$




19:




$19 = 2 cdot 0 + 19$




20:




$20 = 2^0 + 19 = 20! / 19!$




21:




$21 = 20 + 1^9$




22:




$22 = 20 - 1 + sqrt9$




23:




$23 = 20 + 1 cdot sqrt9$




24:




$24 = 2^{0! + 1} cdot (sqrt9)! = 2^{0! + 1}!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$




25:




$25 = (2 + 0!)! + 19$




26:




$26 = 20 + 1 cdot (sqrt9)!$




27:




$27 = 2^{0! + 1}! + sqrt9$




28: Can't get one different from what I've already seen in other answers. Will maybe try later.



29:




$29 = 20 + 1 cdot 9$




30:




$30 = 2^{0! + 1}! + (sqrt9)!$




I know we're supposed to stop at 30, but I accidentally found this fun one:



32:




$32 = sqrt{20!! / (1 + 9)!}$







share|improve this answer































    1














    I wrote a program to determine all representable numbers between 1 and 10000 following the rules, so this should be a comprehensive list.



    0 through 30:




    $$0 = 2times0timesleft(1+9right)$$
    $$1 = 20-19$$
    $$2 = 2+0timesleft(1+9right)$$
    $$3 = 2+0+1^{9}$$
    $$4 = 2-left(0+1-sqrt{9}right)$$
    $$5 = frac{20}{1+sqrt{9}}$$
    $$6 = -2-left(0+1-9right)$$
    $$7 = 2timesleft(0-1right)+9$$
    $$8 = 2times0-left(1-9right)$$
    $$9 = 2times0+1times9$$
    $$10 = 2-left(0+1-9right)$$
    $$11 = 2+0+1times9$$
    $$12 = 2+0+1+9$$
    $$13 = 20-left(1+left(sqrt{9}right)!right)$$
    $$14 = 20-1timesleft(sqrt{9}right)!$$
    $$15 = 20+1-left(sqrt{9}right)!$$
    $$16 = 2timesleft(0-left(1-9right)right)$$
    $$17 = -2+0+19$$
    $$18 = 2timesleft(0+1times9right)$$
    $$19 = 2times0+19$$
    $$20 = 2timesleft(0+1+9right)$$
    $$21 = 2+0+19$$
    $$22 = 20-left(1-sqrt{9}right)$$
    $$23 = 20+1timessqrt{9}$$
    $$24 = 20+1+sqrt{9}$$
    $$25 = 20-left(1-left(sqrt{9}right)!right)$$
    $$26 = 20+1timesleft(sqrt{9}right)!$$
    $$27 = left(2+0+1right)times9$$
    $$28 = 20-left(1-9right)$$
    $$29 = 20+1times9$$
    $$30 = 20+1+9$$




    31 through 100. Interestingly enough, 31 is the smallest number that cannot be done.




    $$32 = sqrt{2^{0+1+9}}$$
    $$33 = left(2+0!+1right)!+9$$
    $$35 = 20+left(-1+left(sqrt{9}right)!right)!!$$
    $$36 = 2timesleft(-0!+19right)$$
    $$38 = 2timesleft(0+19right)$$
    $$39 = 20+19$$
    $$40 = 20timesleft(-1+sqrt{9}right)$$
    $$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt{9}right)!right)$$
    $$42 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)$$
    $$43 = left(left(2+0!right)!right)!!+1-left(sqrt{9}right)!$$
    $$44 = 20+left(1+sqrt{9}right)!$$
    $$45 = left(left(2+0!right)!-1right)times9$$
    $$46 = 2timesleft(-0!+left(1+sqrt{9}right)!right)$$
    $$47 = 2times0-left(1-left(left(sqrt{9}right)!right)!!right)$$
    $$48 = 2timesleft(0+left(1+sqrt{9}right)!right)$$
    $$49 = 2-left(0+1-left(left(sqrt{9}right)!right)!!right)$$
    $$50 = 2timesleft(0!+left(1+sqrt{9}right)!right)$$
    $$51 = 2+0+1+left(left(sqrt{9}right)!right)!!$$
    $$52 = 2+0!+1+left(left(sqrt{9}right)!right)!!$$
    $$53 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
    $$54 = left(2+0!right)!times1times9$$
    $$55 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!!$$
    $$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
    $$57 = left(20-1right)timessqrt{9}$$
    $$58 = left(left(2+0!right)!right)!!+1+9$$
    $$60 = 20times1timessqrt{9}$$
    $$62 = -2+left(0!+1right)^{left(sqrt{9}right)!}$$
    $$63 = left(20+1right)timessqrt{9}$$
    $$64 = 2^{0+1timesleft(sqrt{9}right)!}$$
    $$66 = 2+left(0!+1right)^{left(sqrt{9}right)!}$$
    $$67 = frac{201}{sqrt{9}}$$
    $$68 = 20+1timesleft(left(sqrt{9}right)!right)!!$$
    $$69 = -2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
    $$71 = sqrt{2^{0}+left(1+left(sqrt{9}right)!right)!}$$
    $$72 = left(2+0!right)timesleft(1+sqrt{9}right)!$$
    $$73 = 2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
    $$80 = 20timesleft(1+sqrt{9}right)$$
    $$81 = left(2+0!right)^{1+sqrt{9}}$$
    $$85 = -20+left(1+left(sqrt{9}right)!right)!!$$
    $$90 = frac{left(left(2+0!right)!right)!}{-1+9}$$
    $$92 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
    $$94 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
    $$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt{9}right)!right)!!right)$$
    $$96 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!!right)$$
    $$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt{9}right)!right)!!$$
    $$98 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!!right)$$
    $$99 = left(left(2+0!right)!+1right)!!-left(sqrt{9}right)!$$
    $$100 = 20timesleft(-1+left(sqrt{9}right)!right)$$




    101 through 1000:




    $$102 = left(left(2+0!right)!+1right)!!-sqrt{9}$$
    $$103 = -2+0+left(1+left(sqrt{9}right)!right)!!$$
    $$104 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!!$$
    $$105 = 2times0+left(1+left(sqrt{9}right)!right)!!$$
    $$106 = 2^{0}+left(1+left(sqrt{9}right)!right)!!$$
    $$107 = 2+0+left(1+left(sqrt{9}right)!right)!!$$
    $$108 = 2+0!+left(1+left(sqrt{9}right)!right)!!$$
    $$111 = left(left(2+0!right)!-1right)!-9$$
    $$114 = left(2+0!right)!times19$$
    $$117 = left(left(2+0!right)!-1right)!-sqrt{9}$$
    $$118 = -2+0+left(-1+left(sqrt{9}right)!right)!$$
    $$119 = -left(2^{0}right)+left(-1+left(sqrt{9}right)!right)!$$
    $$120 = 20times1timesleft(sqrt{9}right)!$$
    $$121 = 2^{0}+left(-1+left(sqrt{9}right)!right)!$$
    $$122 = 2+0+left(-1+left(sqrt{9}right)!right)!$$
    $$123 = left(left(2+0!right)!-1right)!+sqrt{9}$$
    $$125 = left(left(2+0!right)!-1right)^{sqrt{9}}$$
    $$126 = left(20+1right)timesleft(sqrt{9}right)!$$
    $$128 = 2^{0+1+left(sqrt{9}right)!}$$
    $$129 = left(left(2+0!right)!-1right)!+9$$
    $$135 = left(left(2+0!right)!-1right)!!times9$$
    $$140 = 20timesleft(1+left(sqrt{9}right)!right)$$
    $$141 = left(left(left(2+0!right)!right)!!-1right)timessqrt{9}$$
    $$142 = 2timessqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
    $$144 = left(2+0!right)!timesleft(1+sqrt{9}right)!$$
    $$147 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
    $$153 = 201-left(left(sqrt{9}right)!right)!!$$
    $$160 = 20timesleft(-1+9right)$$
    $$168 = left(left(2+0!right)!-1right)!+left(left(sqrt{9}right)!right)!!$$
    $$171 = left(20-1right)times9$$
    $$180 = 20times1times9$$
    $$189 = left(20+1right)times9$$
    $$192 = 201-9$$
    $$195 = 201-left(sqrt{9}right)!$$
    $$198 = 201-sqrt{9}$$
    $$200 = 20timesleft(1+9right)$$
    $$204 = 201+sqrt{9}$$
    $$207 = 201+left(sqrt{9}right)!$$
    $$208 = 2timesleft(-0!+left(1+left(sqrt{9}right)!right)!!right)$$
    $$210 = 201+9$$
    $$212 = 2timesleft(0!+left(1+left(sqrt{9}right)!right)!!right)$$
    $$216 = left(2+0!+1right)!times9$$
    $$224 = -left(left(2+0!right)!right)!-left(1-9!!right)$$
    $$225 = -left(left(2+0!right)!right)!+1times9!!$$
    $$226 = -left(left(2+0!right)!right)!+1+9!!$$
    $$238 = 2timesleft(-0!+left(-1+left(sqrt{9}right)!right)!right)$$
    $$240 = 2timesleft(0+left(-1+left(sqrt{9}right)!right)!right)$$
    $$242 = 2timesleft(0!+left(-1+left(sqrt{9}right)!right)!right)$$
    $$243 = left(2+0!right)^{-1+left(sqrt{9}right)!}$$
    $$249 = 201+left(left(sqrt{9}right)!right)!!$$
    $$256 = 2^{0-left(1-9right)}$$
    $$282 = left(left(left(2+0!right)!right)!!-1right)timesleft(sqrt{9}right)!$$
    $$288 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!!$$
    $$294 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
    $$300 = 20timesleft(-1+left(sqrt{9}right)!right)!!$$
    $$315 = left(2+0!right)^{-1}times9!!$$
    $$336 = left(left(2+0!right)!right)!-left(-1+9right)!!$$
    $$343 = left(left(2+0!right)!+1right)^{sqrt{9}}$$
    $$360 = 2^{0-1}timesleft(left(sqrt{9}right)!right)!$$
    $$364 = -20+left(-1+9right)!!$$
    $$375 = left(left(2+0!+1right)!!right)!!-9$$
    $$378 = -left(2+0!right)!+left(-1+9right)!!$$
    $$380 = 20times19$$
    $$381 = -2-left(0!-left(-1+9right)!!right)$$
    $$382 = -2+0+left(-1+9right)!!$$
    $$383 = -left(2^{0}right)+left(-1+9right)!!$$
    $$384 = 2times0+left(-1+9right)!!$$
    $$385 = 2^{0}+left(-1+9right)!!$$
    $$386 = 2+0+left(-1+9right)!!$$
    $$387 = 2+0!+left(-1+9right)!!$$
    $$390 = left(2+0!right)!+left(-1+9right)!!$$
    $$393 = left(left(2+0!+1right)!!right)!!+9$$
    $$400 = 20^{-1+sqrt{9}}$$
    $$404 = 20+left(-1+9right)!!$$
    $$423 = left(left(left(2+0!right)!right)!!-1right)times9$$
    $$432 = left(left(2+0!right)!right)!!times1times9$$
    $$441 = left(left(left(2+0!right)!right)!!+1right)times9$$
    $$472 = 2^{-0!}timesleft(-1+9!!right)$$
    $$473 = 2^{-0!}timesleft(1+9!!right)$$
    $$480 = 20timesleft(1+sqrt{9}right)!$$
    $$504 = left(left(2+0!right)!right)!^{-1}times9!$$
    $$510 = -2+left(0!+1right)^{9}$$
    $$512 = 2^{0+1times9}$$
    $$514 = 2+left(0!+1right)^{9}$$
    $$519 = -201+left(left(sqrt{9}right)!right)!$$
    $$560 = frac{left(left(2+0!right)!+1right)!}{9}$$
    $$561 = -left(left(2+0!+1right)!!right)!!+9!!$$
    $$600 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!$$
    $$603 = 201timessqrt{9}$$
    $$615 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)!!$$
    $$630 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)!!$$
    $$671 = left(left(2+0!right)!right)!-left(1+left(left(sqrt{9}right)!right)!!right)$$
    $$672 = left(left(2+0!right)!right)!-1timesleft(left(sqrt{9}right)!right)!!$$
    $$673 = left(left(2+0!right)!right)!+1-left(left(sqrt{9}right)!right)!!$$
    $$696 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)!$$
    $$699 = -20-left(1-left(left(sqrt{9}right)!right)!right)$$
    $$700 = -20+1timesleft(left(sqrt{9}right)!right)!$$
    $$701 = left(left(2+0!right)!right)!-19$$
    $$705 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!!$$
    $$710 = left(left(2+0!right)!right)!-left(1+9right)$$
    $$711 = left(left(2+0!right)!right)!-1times9$$
    $$712 = left(left(2+0!right)!right)!+1-9$$
    $$713 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)$$
    $$714 = left(left(2+0!right)!right)!-1timesleft(sqrt{9}right)!$$
    $$715 = left(left(2+0!right)!right)!+1-left(sqrt{9}right)!$$
    $$716 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)$$
    $$717 = -2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
    $$718 = 2timesleft(0-1right)+left(left(sqrt{9}right)!right)!$$
    $$719 = 2times0-left(1-left(left(sqrt{9}right)!right)!right)$$
    $$720 = 2times0+1timesleft(left(sqrt{9}right)!right)!$$
    $$721 = 2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
    $$722 = 2+0+1timesleft(left(sqrt{9}right)!right)!$$
    $$723 = 2+0+1+left(left(sqrt{9}right)!right)!$$
    $$724 = 2+0!+1+left(left(sqrt{9}right)!right)!$$
    $$725 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
    $$726 = left(2+0!right)!+1timesleft(left(sqrt{9}right)!right)!$$
    $$727 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!$$
    $$728 = left(left(2+0!right)!right)!-left(1-9right)$$
    $$729 = left(2+0+1right)^{left(sqrt{9}right)!}$$
    $$730 = left(left(2+0!right)!right)!+1+9$$
    $$735 = left(left(2+0!right)!-1right)!!+left(left(sqrt{9}right)!right)!$$
    $$739 = 20-left(1-left(left(sqrt{9}right)!right)!right)$$
    $$740 = 20+1timesleft(left(sqrt{9}right)!right)!$$
    $$741 = 20+1+left(left(sqrt{9}right)!right)!$$
    $$744 = left(left(2+0!right)!right)!+left(1+sqrt{9}right)!$$
    $$766 = 2timesleft(-0!+left(-1+9right)!!right)$$
    $$767 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
    $$768 = 2timesleft(0+left(-1+9right)!!right)$$
    $$769 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!!$$
    $$770 = 2timesleft(0!+left(-1+9right)!!right)$$
    $$825 = -left(left(2+0!right)!-1right)!+9!!$$
    $$840 = frac{left(left(2+0!right)!+1right)!}{left(sqrt{9}right)!}$$
    $$896 = -left(left(2+0!right)!right)!!-left(1-9!!right)$$
    $$897 = -left(left(2+0!right)!right)!!+1times9!!$$
    $$898 = -left(left(2+0!right)!right)!!+1+9!!$$
    $$912 = left(left(2+0!right)!right)!!times19$$
    $$921 = 201+left(left(sqrt{9}right)!right)!$$
    $$924 = -20-left(1-9!!right)$$
    $$925 = -20+1times9!!$$
    $$926 = -20+1+9!!$$
    $$930 = -left(left(2+0!right)!-1right)!!+9!!$$
    $$937 = -left(2+0!+1right)!!+9!!$$
    $$938 = -left(2+0!right)!-left(1-9!!right)$$
    $$939 = -left(2+0!right)!+1times9!!$$
    $$940 = 20timesleft(-1+left(left(sqrt{9}right)!right)!!right)$$
    $$941 = -2-left(0!+1-9!!right)$$
    $$942 = -2-left(0+1-9!!right)$$
    $$943 = 2timesleft(0-1right)+9!!$$
    $$944 = 2times0-left(1-9!!right)$$
    $$945 = 2times0+1times9!!$$
    $$946 = 2-left(0+1-9!!right)$$
    $$947 = 2+0+1times9!!$$
    $$948 = 2+0+1+9!!$$
    $$949 = 2+0!+1+9!!$$
    $$950 = left(2+0!right)!-left(1-9!!right)$$
    $$951 = left(2+0!right)!+1times9!!$$
    $$952 = left(2+0!right)!+1+9!!$$
    $$953 = left(2+0!+1right)!!+9!!$$
    $$960 = 20times1timesleft(left(sqrt{9}right)!right)!!$$
    $$964 = 20-left(1-9!!right)$$
    $$965 = 20+1times9!!$$
    $$966 = 20+1+9!!$$
    $$969 = left(2+0!+1right)!+9!!$$
    $$980 = 20timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
    $$992 = left(left(2+0!right)!right)!!-left(1-9!!right)$$
    $$993 = left(left(2+0!right)!right)!!+1times9!!$$
    $$994 = left(left(2+0!right)!right)!!+1+9!!$$




    1001 through 10000:




    $$1008 = left(20+1right)timesleft(left(sqrt{9}right)!right)!!$$
    $$1024 = 2^{0+1+9}$$
    $$1050 = left(left(2+0!right)!+1right)!!+9!!$$
    $$1065 = left(left(2+0!right)!-1right)!+9!!$$
    $$1080 = left(left(2+0!right)!-1right)!times9$$
    $$1104 = left(left(2+0!right)!right)!+left(-1+9right)!!$$
    $$1146 = 201+9!!$$
    $$1152 = left(2+0!right)timesleft(-1+9right)!!$$
    $$1206 = 201timesleft(sqrt{9}right)!$$
    $$1296 = left(2+0!right)!^{1+sqrt{9}}$$
    $$1329 = left(left(2+0!+1right)!!right)!!+9!!$$
    $$1436 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!right)right)$$
    $$1438 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!right)right)$$
    $$1439 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
    $$1440 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!right)$$
    $$1441 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!$$
    $$1442 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!right)$$
    $$1444 = 2timesleft(0!+1+left(left(sqrt{9}right)!right)!right)$$
    $$1664 = left(left(2+0!right)!right)!-left(1-9!!right)$$
    $$1665 = left(left(2+0!right)!right)!+1times9!!$$
    $$1666 = left(left(2+0!right)!right)!+1+9!!$$
    $$1680 = frac{left(left(2+0!right)!+1right)!}{sqrt{9}}$$
    $$1809 = 201times9$$
    $$1886 = 2timesleft(-0!-left(1-9!!right)right)$$
    $$1888 = 2timesleft(0-left(1-9!!right)right)$$
    $$1890 = 2timesleft(0+1times9!!right)$$
    $$1892 = 2timesleft(0+1+9!!right)$$
    $$1894 = 2timesleft(0!+1+9!!right)$$
    $$1920 = 2^{-0!}timesleft(1+9right)!!$$
    $$2019 = 2019$$
    $$2048 = 2^{0!+1+9}$$
    $$2100 = 20timesleft(1+left(sqrt{9}right)!right)!!$$
    $$2145 = frac{left(left(left(2+0!right)!-1right)!!right)!!}{9!!}$$
    $$2157 = left(left(left(2+0!right)!right)!-1right)timessqrt{9}$$
    $$2160 = left(2+0+1right)timesleft(left(sqrt{9}right)!right)!$$
    $$2163 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!right)$$
    $$2187 = left(2+0!right)^{1+left(sqrt{9}right)!}$$
    $$2256 = left(left(left(2+0!right)!right)!!-1right)timesleft(left(sqrt{9}right)!right)!!$$
    $$2304 = left(2+0!right)!timesleft(-1+9right)!!$$
    $$2352 = left(left(2+0!right)!right)!!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
    $$2400 = 20timesleft(-1+left(sqrt{9}right)!right)!$$
    $$2520 = 2^{-0!}timesleft(1+left(sqrt{9}right)!right)!$$
    $$2832 = left(2+0!right)timesleft(-1+9!!right)$$
    $$2835 = left(2+0+1right)times9!!$$
    $$2838 = left(2+0!right)timesleft(1+9!!right)$$
    $$2880 = 2timesleft(0!+1right)timesleft(left(sqrt{9}right)!right)!$$
    $$3120 = -left(left(2+0!right)!right)!+left(1+9right)!!$$
    $$3375 = left(left(2+0!right)!-1right)!!^{sqrt{9}}$$
    $$3456 = left(left(2+0!+1right)!!right)!!times9$$
    $$3600 = left(left(2+0!right)!-1right)timesleft(left(sqrt{9}right)!right)!$$
    $$3780 = 2timesleft(0!+1right)times9!!$$
    $$3792 = -left(left(2+0!right)!right)!!+left(1+9right)!!$$
    $$3820 = -20+left(1+9right)!!$$
    $$3834 = -left(2+0!right)!+left(1+9right)!!$$
    $$3837 = -2-left(0!-left(1+9right)!!right)$$
    $$3838 = -2+0+left(1+9right)!!$$
    $$3839 = -left(2^{0}right)+left(1+9right)!!$$
    $$3840 = 2times0+left(1+9right)!!$$
    $$3841 = 2^{0}+left(1+9right)!!$$
    $$3842 = 2+0+left(1+9right)!!$$
    $$3843 = 2+0!+left(1+9right)!!$$
    $$3846 = left(2+0!right)!+left(1+9right)!!$$
    $$3860 = 20+left(1+9right)!!$$
    $$3888 = left(left(2+0!right)!right)!!+left(1+9right)!!$$
    $$4094 = -2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
    $$4095 = left(left(2+0!right)!+1right)!-9!!$$
    $$4096 = 2^{left(0!+1right)timesleft(sqrt{9}right)!}$$
    $$4098 = 2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
    $$4314 = left(left(left(2+0!right)!right)!-1right)timesleft(sqrt{9}right)!$$
    $$4320 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!$$
    $$4326 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!right)$$
    $$4480 = frac{left(left(2+0!+1right)!!right)!}{9}$$
    $$4560 = left(left(2+0!right)!right)!+left(1+9right)!!$$
    $$4725 = left(left(2+0!right)!-1right)times9!!$$
    $$4992 = left(left(2+0!right)!+1right)!-left(left(sqrt{9}right)!right)!!$$
    $$5020 = -20+left(1+left(sqrt{9}right)!right)!$$
    $$5031 = left(left(2+0!right)!+1right)!-9$$
    $$5034 = left(left(2+0!right)!+1right)!-left(sqrt{9}right)!$$
    $$5037 = left(left(2+0!right)!+1right)!-sqrt{9}$$
    $$5038 = -2+0+left(1+left(sqrt{9}right)!right)!$$
    $$5039 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!$$
    $$5040 = 2times0+left(1+left(sqrt{9}right)!right)!$$
    $$5041 = 2^{0}+left(1+left(sqrt{9}right)!right)!$$
    $$5042 = 2+0+left(1+left(sqrt{9}right)!right)!$$
    $$5043 = 2+0!+left(1+left(sqrt{9}right)!right)!$$
    $$5046 = left(2+0!right)!+left(1+left(sqrt{9}right)!right)!$$
    $$5049 = left(left(2+0!right)!+1right)!+9$$
    $$5060 = 20+left(1+left(sqrt{9}right)!right)!$$
    $$5088 = left(left(2+0!right)!right)!!+left(1+left(sqrt{9}right)!right)!$$
    $$5664 = left(2+0!right)!timesleft(-1+9!!right)$$
    $$5670 = left(2+0!right)!times1times9!!$$
    $$5676 = left(2+0!right)!timesleft(1+9!!right)$$
    $$5760 = left(left(2+0!right)!right)!timesleft(-1+9right)$$
    $$5985 = left(left(2+0!right)!+1right)!+9!!$$
    $$6471 = left(left(left(2+0!right)!right)!-1right)times9$$
    $$6480 = left(left(2+0!right)!right)!times1times9$$
    $$6489 = left(left(left(2+0!right)!right)!+1right)times9$$
    $$6561 = left(2+0!right)^{-1+9}$$
    $$6615 = left(left(2+0!right)!+1right)times9!!$$
    $$6720 = frac{left(left(2+0!+1right)!!right)!}{left(sqrt{9}right)!}$$
    $$6859 = left(20-1right)^{sqrt{9}}$$
    $$7200 = left(left(2+0!right)!right)!timesleft(1+9right)$$
    $$7560 = left(2+0!+1right)!!times9!!$$
    $$7678 = 2timesleft(-0!+left(1+9right)!!right)$$
    $$7680 = 20timesleft(-1+9right)!!$$
    $$7682 = 2timesleft(0!+left(1+9right)!!right)$$
    $$7776 = left(2+0!right)!^{-1+left(sqrt{9}right)!}$$
    $$8000 = 20^{1timessqrt{9}}$$
    $$8192 = 2timessqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
    $$9261 = left(20+1right)^{sqrt{9}}$$
    $$9648 = 201timesleft(left(sqrt{9}right)!right)!!$$






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      4 Answers
      4






      active

      oldest

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      4 Answers
      4






      active

      oldest

      votes









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      votes









      3














      1




      1 = 2^(0*19)




      2




      2 = 2 + (0*19)




      3




      3 = 2 + 0!^19




      4




      4 = 2 ^ (0! + 1 ^ 9)




      5




      -((2 + 0! + 1) - 9)




      6




      -((2 + 0 + 1) - 9))




      7




      -((2 + 0*1 - 9))




      8




      -((2 - 01) - 9)




      9




      2*0*1 + 9




      10




      2*0 + 1 + 9




      11




      2 + 0*1 + 9




      12




      2 + 0 + 1 + 9




      13




      2 + 0! + 1 + 9




      14




      (2 + 0!)! - 1 + 9




      15




      (2 + 0 + 1)! + 9




      16




      (2 + 0!)! + 1 + 9




      17




      20 - (1 * sqrt(9))




      18




      (2 + (0 * 1)) * 9




      19




      20 - 1^9




      20




      20 * 1^9




      21




      20 + 1^9




      22




      2 + 0! + 19




      23




      20 + 1 * sqrt(9)




      24




      20 + 1 + sqrt(9)




      25




      2 || (0! + 1 + sqrt (9))




      explained:




      Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.




      26




      2 || ((0! + 1) * sqrt(9))




      27




      (2 + 0 + 1) * 9




      28




      ((2 + 0!) || 1) - sqrt(9)




      29




      (2 + (0 * 1)) || 9




      30




      20 + 1 + 9







      share|improve this answer























      • edited for clarity
        – flashstorm
        10 hours ago










      • @flashstorm Um... $3ne 2+0^{19}$
        – Frpzzd
        10 hours ago






      • 1




        was missing an !
        – flashstorm
        10 hours ago










      • Ding! Fries are done :)
        – flashstorm
        10 hours ago
















      3














      1




      1 = 2^(0*19)




      2




      2 = 2 + (0*19)




      3




      3 = 2 + 0!^19




      4




      4 = 2 ^ (0! + 1 ^ 9)




      5




      -((2 + 0! + 1) - 9)




      6




      -((2 + 0 + 1) - 9))




      7




      -((2 + 0*1 - 9))




      8




      -((2 - 01) - 9)




      9




      2*0*1 + 9




      10




      2*0 + 1 + 9




      11




      2 + 0*1 + 9




      12




      2 + 0 + 1 + 9




      13




      2 + 0! + 1 + 9




      14




      (2 + 0!)! - 1 + 9




      15




      (2 + 0 + 1)! + 9




      16




      (2 + 0!)! + 1 + 9




      17




      20 - (1 * sqrt(9))




      18




      (2 + (0 * 1)) * 9




      19




      20 - 1^9




      20




      20 * 1^9




      21




      20 + 1^9




      22




      2 + 0! + 19




      23




      20 + 1 * sqrt(9)




      24




      20 + 1 + sqrt(9)




      25




      2 || (0! + 1 + sqrt (9))




      explained:




      Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.




      26




      2 || ((0! + 1) * sqrt(9))




      27




      (2 + 0 + 1) * 9




      28




      ((2 + 0!) || 1) - sqrt(9)




      29




      (2 + (0 * 1)) || 9




      30




      20 + 1 + 9







      share|improve this answer























      • edited for clarity
        – flashstorm
        10 hours ago










      • @flashstorm Um... $3ne 2+0^{19}$
        – Frpzzd
        10 hours ago






      • 1




        was missing an !
        – flashstorm
        10 hours ago










      • Ding! Fries are done :)
        – flashstorm
        10 hours ago














      3












      3








      3






      1




      1 = 2^(0*19)




      2




      2 = 2 + (0*19)




      3




      3 = 2 + 0!^19




      4




      4 = 2 ^ (0! + 1 ^ 9)




      5




      -((2 + 0! + 1) - 9)




      6




      -((2 + 0 + 1) - 9))




      7




      -((2 + 0*1 - 9))




      8




      -((2 - 01) - 9)




      9




      2*0*1 + 9




      10




      2*0 + 1 + 9




      11




      2 + 0*1 + 9




      12




      2 + 0 + 1 + 9




      13




      2 + 0! + 1 + 9




      14




      (2 + 0!)! - 1 + 9




      15




      (2 + 0 + 1)! + 9




      16




      (2 + 0!)! + 1 + 9




      17




      20 - (1 * sqrt(9))




      18




      (2 + (0 * 1)) * 9




      19




      20 - 1^9




      20




      20 * 1^9




      21




      20 + 1^9




      22




      2 + 0! + 19




      23




      20 + 1 * sqrt(9)




      24




      20 + 1 + sqrt(9)




      25




      2 || (0! + 1 + sqrt (9))




      explained:




      Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.




      26




      2 || ((0! + 1) * sqrt(9))




      27




      (2 + 0 + 1) * 9




      28




      ((2 + 0!) || 1) - sqrt(9)




      29




      (2 + (0 * 1)) || 9




      30




      20 + 1 + 9







      share|improve this answer














      1




      1 = 2^(0*19)




      2




      2 = 2 + (0*19)




      3




      3 = 2 + 0!^19




      4




      4 = 2 ^ (0! + 1 ^ 9)




      5




      -((2 + 0! + 1) - 9)




      6




      -((2 + 0 + 1) - 9))




      7




      -((2 + 0*1 - 9))




      8




      -((2 - 01) - 9)




      9




      2*0*1 + 9




      10




      2*0 + 1 + 9




      11




      2 + 0*1 + 9




      12




      2 + 0 + 1 + 9




      13




      2 + 0! + 1 + 9




      14




      (2 + 0!)! - 1 + 9




      15




      (2 + 0 + 1)! + 9




      16




      (2 + 0!)! + 1 + 9




      17




      20 - (1 * sqrt(9))




      18




      (2 + (0 * 1)) * 9




      19




      20 - 1^9




      20




      20 * 1^9




      21




      20 + 1^9




      22




      2 + 0! + 19




      23




      20 + 1 * sqrt(9)




      24




      20 + 1 + sqrt(9)




      25




      2 || (0! + 1 + sqrt (9))




      explained:




      Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.




      26




      2 || ((0! + 1) * sqrt(9))




      27




      (2 + 0 + 1) * 9




      28




      ((2 + 0!) || 1) - sqrt(9)




      29




      (2 + (0 * 1)) || 9




      30




      20 + 1 + 9








      share|improve this answer














      share|improve this answer



      share|improve this answer








      edited 10 hours ago

























      answered 11 hours ago









      flashstorm

      7329




      7329












      • edited for clarity
        – flashstorm
        10 hours ago










      • @flashstorm Um... $3ne 2+0^{19}$
        – Frpzzd
        10 hours ago






      • 1




        was missing an !
        – flashstorm
        10 hours ago










      • Ding! Fries are done :)
        – flashstorm
        10 hours ago


















      • edited for clarity
        – flashstorm
        10 hours ago










      • @flashstorm Um... $3ne 2+0^{19}$
        – Frpzzd
        10 hours ago






      • 1




        was missing an !
        – flashstorm
        10 hours ago










      • Ding! Fries are done :)
        – flashstorm
        10 hours ago
















      edited for clarity
      – flashstorm
      10 hours ago




      edited for clarity
      – flashstorm
      10 hours ago












      @flashstorm Um... $3ne 2+0^{19}$
      – Frpzzd
      10 hours ago




      @flashstorm Um... $3ne 2+0^{19}$
      – Frpzzd
      10 hours ago




      1




      1




      was missing an !
      – flashstorm
      10 hours ago




      was missing an !
      – flashstorm
      10 hours ago












      Ding! Fries are done :)
      – flashstorm
      10 hours ago




      Ding! Fries are done :)
      – flashstorm
      10 hours ago











      6















      $$1=20-19$$
      $$2=2+0cdot 19=20div(1+9)$$
      $$3=2cdot 0cdot 1+sqrt{9}=-(2+0+1)!+9$$
      $$4=2^{0-1+sqrt{9}}$$
      $$5=20div (1+sqrt{9})$$
      $$6=(2cdot 0cdot 1+sqrt{9})!$$
      $$7=-2-0cdot 1+9$$
      $$8=2^{0cdot 1+sqrt{9}}$$
      $$9=2cdot 0cdot 1+9$$
      $$10=2cdot 0+1+9=20-1-9$$
      $$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
      $$12=2+0+1+9=20+1-9$$
      $$13=2+0!+1+9=2^{0!+1}+9$$
      $$14=(2+0!)!-1+9$$
      $$15=(2+0+1)!+9$$
      $$16=2^{0+1+sqrt{9}}$$
      $$17=20-sqrt{1cdot 9}$$
      $$18=20+1-sqrt{9}$$
      $$19=2cdot 0+19$$
      $$20=2^0+19$$
      $$21=20+1^9$$
      $$22=2cdot (0!+1+9)$$
      $$23=20+sqrt{1cdot 9}$$
      $$24=2^{0-1+sqrt{9}}!=20+1+sqrt{9}$$
      $$25=(2+0!)!+19$$
      $$26=2+0+(1+sqrt{9})!$$
      $$27=(2+0+1)^{sqrt{9}}$$
      $$28=20-1+9$$
      $$29=20+1cdot 9=20cdot 1+9$$
      $$30=20+1+9$$




      DONE!






      share|improve this answer























      • All finished now! :D
        – Frpzzd
        10 hours ago










      • Great :) But Spoiler-Tags would be nice ;)
        – André
        8 hours ago












      • @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
        – Frpzzd
        8 hours ago






      • 1




        @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
        – Frpzzd
        8 hours ago










      • Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
        – user1207177
        8 hours ago


















      6















      $$1=20-19$$
      $$2=2+0cdot 19=20div(1+9)$$
      $$3=2cdot 0cdot 1+sqrt{9}=-(2+0+1)!+9$$
      $$4=2^{0-1+sqrt{9}}$$
      $$5=20div (1+sqrt{9})$$
      $$6=(2cdot 0cdot 1+sqrt{9})!$$
      $$7=-2-0cdot 1+9$$
      $$8=2^{0cdot 1+sqrt{9}}$$
      $$9=2cdot 0cdot 1+9$$
      $$10=2cdot 0+1+9=20-1-9$$
      $$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
      $$12=2+0+1+9=20+1-9$$
      $$13=2+0!+1+9=2^{0!+1}+9$$
      $$14=(2+0!)!-1+9$$
      $$15=(2+0+1)!+9$$
      $$16=2^{0+1+sqrt{9}}$$
      $$17=20-sqrt{1cdot 9}$$
      $$18=20+1-sqrt{9}$$
      $$19=2cdot 0+19$$
      $$20=2^0+19$$
      $$21=20+1^9$$
      $$22=2cdot (0!+1+9)$$
      $$23=20+sqrt{1cdot 9}$$
      $$24=2^{0-1+sqrt{9}}!=20+1+sqrt{9}$$
      $$25=(2+0!)!+19$$
      $$26=2+0+(1+sqrt{9})!$$
      $$27=(2+0+1)^{sqrt{9}}$$
      $$28=20-1+9$$
      $$29=20+1cdot 9=20cdot 1+9$$
      $$30=20+1+9$$




      DONE!






      share|improve this answer























      • All finished now! :D
        – Frpzzd
        10 hours ago










      • Great :) But Spoiler-Tags would be nice ;)
        – André
        8 hours ago












      • @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
        – Frpzzd
        8 hours ago






      • 1




        @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
        – Frpzzd
        8 hours ago










      • Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
        – user1207177
        8 hours ago
















      6












      6








      6







      $$1=20-19$$
      $$2=2+0cdot 19=20div(1+9)$$
      $$3=2cdot 0cdot 1+sqrt{9}=-(2+0+1)!+9$$
      $$4=2^{0-1+sqrt{9}}$$
      $$5=20div (1+sqrt{9})$$
      $$6=(2cdot 0cdot 1+sqrt{9})!$$
      $$7=-2-0cdot 1+9$$
      $$8=2^{0cdot 1+sqrt{9}}$$
      $$9=2cdot 0cdot 1+9$$
      $$10=2cdot 0+1+9=20-1-9$$
      $$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
      $$12=2+0+1+9=20+1-9$$
      $$13=2+0!+1+9=2^{0!+1}+9$$
      $$14=(2+0!)!-1+9$$
      $$15=(2+0+1)!+9$$
      $$16=2^{0+1+sqrt{9}}$$
      $$17=20-sqrt{1cdot 9}$$
      $$18=20+1-sqrt{9}$$
      $$19=2cdot 0+19$$
      $$20=2^0+19$$
      $$21=20+1^9$$
      $$22=2cdot (0!+1+9)$$
      $$23=20+sqrt{1cdot 9}$$
      $$24=2^{0-1+sqrt{9}}!=20+1+sqrt{9}$$
      $$25=(2+0!)!+19$$
      $$26=2+0+(1+sqrt{9})!$$
      $$27=(2+0+1)^{sqrt{9}}$$
      $$28=20-1+9$$
      $$29=20+1cdot 9=20cdot 1+9$$
      $$30=20+1+9$$




      DONE!






      share|improve this answer















      $$1=20-19$$
      $$2=2+0cdot 19=20div(1+9)$$
      $$3=2cdot 0cdot 1+sqrt{9}=-(2+0+1)!+9$$
      $$4=2^{0-1+sqrt{9}}$$
      $$5=20div (1+sqrt{9})$$
      $$6=(2cdot 0cdot 1+sqrt{9})!$$
      $$7=-2-0cdot 1+9$$
      $$8=2^{0cdot 1+sqrt{9}}$$
      $$9=2cdot 0cdot 1+9$$
      $$10=2cdot 0+1+9=20-1-9$$
      $$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
      $$12=2+0+1+9=20+1-9$$
      $$13=2+0!+1+9=2^{0!+1}+9$$
      $$14=(2+0!)!-1+9$$
      $$15=(2+0+1)!+9$$
      $$16=2^{0+1+sqrt{9}}$$
      $$17=20-sqrt{1cdot 9}$$
      $$18=20+1-sqrt{9}$$
      $$19=2cdot 0+19$$
      $$20=2^0+19$$
      $$21=20+1^9$$
      $$22=2cdot (0!+1+9)$$
      $$23=20+sqrt{1cdot 9}$$
      $$24=2^{0-1+sqrt{9}}!=20+1+sqrt{9}$$
      $$25=(2+0!)!+19$$
      $$26=2+0+(1+sqrt{9})!$$
      $$27=(2+0+1)^{sqrt{9}}$$
      $$28=20-1+9$$
      $$29=20+1cdot 9=20cdot 1+9$$
      $$30=20+1+9$$




      DONE!







      share|improve this answer














      share|improve this answer



      share|improve this answer








      edited 2 hours ago









      Hugh

      1,3631617




      1,3631617










      answered 10 hours ago









      Frpzzd

      841120




      841120












      • All finished now! :D
        – Frpzzd
        10 hours ago










      • Great :) But Spoiler-Tags would be nice ;)
        – André
        8 hours ago












      • @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
        – Frpzzd
        8 hours ago






      • 1




        @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
        – Frpzzd
        8 hours ago










      • Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
        – user1207177
        8 hours ago




















      • All finished now! :D
        – Frpzzd
        10 hours ago










      • Great :) But Spoiler-Tags would be nice ;)
        – André
        8 hours ago












      • @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
        – Frpzzd
        8 hours ago






      • 1




        @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
        – Frpzzd
        8 hours ago










      • Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
        – user1207177
        8 hours ago


















      All finished now! :D
      – Frpzzd
      10 hours ago




      All finished now! :D
      – Frpzzd
      10 hours ago












      Great :) But Spoiler-Tags would be nice ;)
      – André
      8 hours ago






      Great :) But Spoiler-Tags would be nice ;)
      – André
      8 hours ago














      @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
      – Frpzzd
      8 hours ago




      @André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
      – Frpzzd
      8 hours ago




      1




      1




      @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
      – Frpzzd
      8 hours ago




      @André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
      – Frpzzd
      8 hours ago












      Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
      – user1207177
      8 hours ago






      Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
      – user1207177
      8 hours ago













      2














      1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.



      8:




      $8 = sqrt{(2^{0!+1}!!)!! / (sqrt9)!}$




      16:




      $16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$




      17:




      $17 = (2 + 0! + 1)!! + 9$




      18:




      $18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^{0+1} cdot 9$




      19:




      $19 = 2 cdot 0 + 19$




      20:




      $20 = 2^0 + 19 = 20! / 19!$




      21:




      $21 = 20 + 1^9$




      22:




      $22 = 20 - 1 + sqrt9$




      23:




      $23 = 20 + 1 cdot sqrt9$




      24:




      $24 = 2^{0! + 1} cdot (sqrt9)! = 2^{0! + 1}!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$




      25:




      $25 = (2 + 0!)! + 19$




      26:




      $26 = 20 + 1 cdot (sqrt9)!$




      27:




      $27 = 2^{0! + 1}! + sqrt9$




      28: Can't get one different from what I've already seen in other answers. Will maybe try later.



      29:




      $29 = 20 + 1 cdot 9$




      30:




      $30 = 2^{0! + 1}! + (sqrt9)!$




      I know we're supposed to stop at 30, but I accidentally found this fun one:



      32:




      $32 = sqrt{20!! / (1 + 9)!}$







      share|improve this answer




























        2














        1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.



        8:




        $8 = sqrt{(2^{0!+1}!!)!! / (sqrt9)!}$




        16:




        $16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$




        17:




        $17 = (2 + 0! + 1)!! + 9$




        18:




        $18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^{0+1} cdot 9$




        19:




        $19 = 2 cdot 0 + 19$




        20:




        $20 = 2^0 + 19 = 20! / 19!$




        21:




        $21 = 20 + 1^9$




        22:




        $22 = 20 - 1 + sqrt9$




        23:




        $23 = 20 + 1 cdot sqrt9$




        24:




        $24 = 2^{0! + 1} cdot (sqrt9)! = 2^{0! + 1}!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$




        25:




        $25 = (2 + 0!)! + 19$




        26:




        $26 = 20 + 1 cdot (sqrt9)!$




        27:




        $27 = 2^{0! + 1}! + sqrt9$




        28: Can't get one different from what I've already seen in other answers. Will maybe try later.



        29:




        $29 = 20 + 1 cdot 9$




        30:




        $30 = 2^{0! + 1}! + (sqrt9)!$




        I know we're supposed to stop at 30, but I accidentally found this fun one:



        32:




        $32 = sqrt{20!! / (1 + 9)!}$







        share|improve this answer


























          2












          2








          2






          1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.



          8:




          $8 = sqrt{(2^{0!+1}!!)!! / (sqrt9)!}$




          16:




          $16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$




          17:




          $17 = (2 + 0! + 1)!! + 9$




          18:




          $18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^{0+1} cdot 9$




          19:




          $19 = 2 cdot 0 + 19$




          20:




          $20 = 2^0 + 19 = 20! / 19!$




          21:




          $21 = 20 + 1^9$




          22:




          $22 = 20 - 1 + sqrt9$




          23:




          $23 = 20 + 1 cdot sqrt9$




          24:




          $24 = 2^{0! + 1} cdot (sqrt9)! = 2^{0! + 1}!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$




          25:




          $25 = (2 + 0!)! + 19$




          26:




          $26 = 20 + 1 cdot (sqrt9)!$




          27:




          $27 = 2^{0! + 1}! + sqrt9$




          28: Can't get one different from what I've already seen in other answers. Will maybe try later.



          29:




          $29 = 20 + 1 cdot 9$




          30:




          $30 = 2^{0! + 1}! + (sqrt9)!$




          I know we're supposed to stop at 30, but I accidentally found this fun one:



          32:




          $32 = sqrt{20!! / (1 + 9)!}$







          share|improve this answer














          1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.



          8:




          $8 = sqrt{(2^{0!+1}!!)!! / (sqrt9)!}$




          16:




          $16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$




          17:




          $17 = (2 + 0! + 1)!! + 9$




          18:




          $18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^{0+1} cdot 9$




          19:




          $19 = 2 cdot 0 + 19$




          20:




          $20 = 2^0 + 19 = 20! / 19!$




          21:




          $21 = 20 + 1^9$




          22:




          $22 = 20 - 1 + sqrt9$




          23:




          $23 = 20 + 1 cdot sqrt9$




          24:




          $24 = 2^{0! + 1} cdot (sqrt9)! = 2^{0! + 1}!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$




          25:




          $25 = (2 + 0!)! + 19$




          26:




          $26 = 20 + 1 cdot (sqrt9)!$




          27:




          $27 = 2^{0! + 1}! + sqrt9$




          28: Can't get one different from what I've already seen in other answers. Will maybe try later.



          29:




          $29 = 20 + 1 cdot 9$




          30:




          $30 = 2^{0! + 1}! + (sqrt9)!$




          I know we're supposed to stop at 30, but I accidentally found this fun one:



          32:




          $32 = sqrt{20!! / (1 + 9)!}$








          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 9 hours ago

























          answered 10 hours ago









          tilper

          872514




          872514























              1














              I wrote a program to determine all representable numbers between 1 and 10000 following the rules, so this should be a comprehensive list.



              0 through 30:




              $$0 = 2times0timesleft(1+9right)$$
              $$1 = 20-19$$
              $$2 = 2+0timesleft(1+9right)$$
              $$3 = 2+0+1^{9}$$
              $$4 = 2-left(0+1-sqrt{9}right)$$
              $$5 = frac{20}{1+sqrt{9}}$$
              $$6 = -2-left(0+1-9right)$$
              $$7 = 2timesleft(0-1right)+9$$
              $$8 = 2times0-left(1-9right)$$
              $$9 = 2times0+1times9$$
              $$10 = 2-left(0+1-9right)$$
              $$11 = 2+0+1times9$$
              $$12 = 2+0+1+9$$
              $$13 = 20-left(1+left(sqrt{9}right)!right)$$
              $$14 = 20-1timesleft(sqrt{9}right)!$$
              $$15 = 20+1-left(sqrt{9}right)!$$
              $$16 = 2timesleft(0-left(1-9right)right)$$
              $$17 = -2+0+19$$
              $$18 = 2timesleft(0+1times9right)$$
              $$19 = 2times0+19$$
              $$20 = 2timesleft(0+1+9right)$$
              $$21 = 2+0+19$$
              $$22 = 20-left(1-sqrt{9}right)$$
              $$23 = 20+1timessqrt{9}$$
              $$24 = 20+1+sqrt{9}$$
              $$25 = 20-left(1-left(sqrt{9}right)!right)$$
              $$26 = 20+1timesleft(sqrt{9}right)!$$
              $$27 = left(2+0+1right)times9$$
              $$28 = 20-left(1-9right)$$
              $$29 = 20+1times9$$
              $$30 = 20+1+9$$




              31 through 100. Interestingly enough, 31 is the smallest number that cannot be done.




              $$32 = sqrt{2^{0+1+9}}$$
              $$33 = left(2+0!+1right)!+9$$
              $$35 = 20+left(-1+left(sqrt{9}right)!right)!!$$
              $$36 = 2timesleft(-0!+19right)$$
              $$38 = 2timesleft(0+19right)$$
              $$39 = 20+19$$
              $$40 = 20timesleft(-1+sqrt{9}right)$$
              $$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt{9}right)!right)$$
              $$42 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)$$
              $$43 = left(left(2+0!right)!right)!!+1-left(sqrt{9}right)!$$
              $$44 = 20+left(1+sqrt{9}right)!$$
              $$45 = left(left(2+0!right)!-1right)times9$$
              $$46 = 2timesleft(-0!+left(1+sqrt{9}right)!right)$$
              $$47 = 2times0-left(1-left(left(sqrt{9}right)!right)!!right)$$
              $$48 = 2timesleft(0+left(1+sqrt{9}right)!right)$$
              $$49 = 2-left(0+1-left(left(sqrt{9}right)!right)!!right)$$
              $$50 = 2timesleft(0!+left(1+sqrt{9}right)!right)$$
              $$51 = 2+0+1+left(left(sqrt{9}right)!right)!!$$
              $$52 = 2+0!+1+left(left(sqrt{9}right)!right)!!$$
              $$53 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
              $$54 = left(2+0!right)!times1times9$$
              $$55 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!!$$
              $$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
              $$57 = left(20-1right)timessqrt{9}$$
              $$58 = left(left(2+0!right)!right)!!+1+9$$
              $$60 = 20times1timessqrt{9}$$
              $$62 = -2+left(0!+1right)^{left(sqrt{9}right)!}$$
              $$63 = left(20+1right)timessqrt{9}$$
              $$64 = 2^{0+1timesleft(sqrt{9}right)!}$$
              $$66 = 2+left(0!+1right)^{left(sqrt{9}right)!}$$
              $$67 = frac{201}{sqrt{9}}$$
              $$68 = 20+1timesleft(left(sqrt{9}right)!right)!!$$
              $$69 = -2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
              $$71 = sqrt{2^{0}+left(1+left(sqrt{9}right)!right)!}$$
              $$72 = left(2+0!right)timesleft(1+sqrt{9}right)!$$
              $$73 = 2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
              $$80 = 20timesleft(1+sqrt{9}right)$$
              $$81 = left(2+0!right)^{1+sqrt{9}}$$
              $$85 = -20+left(1+left(sqrt{9}right)!right)!!$$
              $$90 = frac{left(left(2+0!right)!right)!}{-1+9}$$
              $$92 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
              $$94 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
              $$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt{9}right)!right)!!right)$$
              $$96 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!!right)$$
              $$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt{9}right)!right)!!$$
              $$98 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!!right)$$
              $$99 = left(left(2+0!right)!+1right)!!-left(sqrt{9}right)!$$
              $$100 = 20timesleft(-1+left(sqrt{9}right)!right)$$




              101 through 1000:




              $$102 = left(left(2+0!right)!+1right)!!-sqrt{9}$$
              $$103 = -2+0+left(1+left(sqrt{9}right)!right)!!$$
              $$104 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!!$$
              $$105 = 2times0+left(1+left(sqrt{9}right)!right)!!$$
              $$106 = 2^{0}+left(1+left(sqrt{9}right)!right)!!$$
              $$107 = 2+0+left(1+left(sqrt{9}right)!right)!!$$
              $$108 = 2+0!+left(1+left(sqrt{9}right)!right)!!$$
              $$111 = left(left(2+0!right)!-1right)!-9$$
              $$114 = left(2+0!right)!times19$$
              $$117 = left(left(2+0!right)!-1right)!-sqrt{9}$$
              $$118 = -2+0+left(-1+left(sqrt{9}right)!right)!$$
              $$119 = -left(2^{0}right)+left(-1+left(sqrt{9}right)!right)!$$
              $$120 = 20times1timesleft(sqrt{9}right)!$$
              $$121 = 2^{0}+left(-1+left(sqrt{9}right)!right)!$$
              $$122 = 2+0+left(-1+left(sqrt{9}right)!right)!$$
              $$123 = left(left(2+0!right)!-1right)!+sqrt{9}$$
              $$125 = left(left(2+0!right)!-1right)^{sqrt{9}}$$
              $$126 = left(20+1right)timesleft(sqrt{9}right)!$$
              $$128 = 2^{0+1+left(sqrt{9}right)!}$$
              $$129 = left(left(2+0!right)!-1right)!+9$$
              $$135 = left(left(2+0!right)!-1right)!!times9$$
              $$140 = 20timesleft(1+left(sqrt{9}right)!right)$$
              $$141 = left(left(left(2+0!right)!right)!!-1right)timessqrt{9}$$
              $$142 = 2timessqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
              $$144 = left(2+0!right)!timesleft(1+sqrt{9}right)!$$
              $$147 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
              $$153 = 201-left(left(sqrt{9}right)!right)!!$$
              $$160 = 20timesleft(-1+9right)$$
              $$168 = left(left(2+0!right)!-1right)!+left(left(sqrt{9}right)!right)!!$$
              $$171 = left(20-1right)times9$$
              $$180 = 20times1times9$$
              $$189 = left(20+1right)times9$$
              $$192 = 201-9$$
              $$195 = 201-left(sqrt{9}right)!$$
              $$198 = 201-sqrt{9}$$
              $$200 = 20timesleft(1+9right)$$
              $$204 = 201+sqrt{9}$$
              $$207 = 201+left(sqrt{9}right)!$$
              $$208 = 2timesleft(-0!+left(1+left(sqrt{9}right)!right)!!right)$$
              $$210 = 201+9$$
              $$212 = 2timesleft(0!+left(1+left(sqrt{9}right)!right)!!right)$$
              $$216 = left(2+0!+1right)!times9$$
              $$224 = -left(left(2+0!right)!right)!-left(1-9!!right)$$
              $$225 = -left(left(2+0!right)!right)!+1times9!!$$
              $$226 = -left(left(2+0!right)!right)!+1+9!!$$
              $$238 = 2timesleft(-0!+left(-1+left(sqrt{9}right)!right)!right)$$
              $$240 = 2timesleft(0+left(-1+left(sqrt{9}right)!right)!right)$$
              $$242 = 2timesleft(0!+left(-1+left(sqrt{9}right)!right)!right)$$
              $$243 = left(2+0!right)^{-1+left(sqrt{9}right)!}$$
              $$249 = 201+left(left(sqrt{9}right)!right)!!$$
              $$256 = 2^{0-left(1-9right)}$$
              $$282 = left(left(left(2+0!right)!right)!!-1right)timesleft(sqrt{9}right)!$$
              $$288 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!!$$
              $$294 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
              $$300 = 20timesleft(-1+left(sqrt{9}right)!right)!!$$
              $$315 = left(2+0!right)^{-1}times9!!$$
              $$336 = left(left(2+0!right)!right)!-left(-1+9right)!!$$
              $$343 = left(left(2+0!right)!+1right)^{sqrt{9}}$$
              $$360 = 2^{0-1}timesleft(left(sqrt{9}right)!right)!$$
              $$364 = -20+left(-1+9right)!!$$
              $$375 = left(left(2+0!+1right)!!right)!!-9$$
              $$378 = -left(2+0!right)!+left(-1+9right)!!$$
              $$380 = 20times19$$
              $$381 = -2-left(0!-left(-1+9right)!!right)$$
              $$382 = -2+0+left(-1+9right)!!$$
              $$383 = -left(2^{0}right)+left(-1+9right)!!$$
              $$384 = 2times0+left(-1+9right)!!$$
              $$385 = 2^{0}+left(-1+9right)!!$$
              $$386 = 2+0+left(-1+9right)!!$$
              $$387 = 2+0!+left(-1+9right)!!$$
              $$390 = left(2+0!right)!+left(-1+9right)!!$$
              $$393 = left(left(2+0!+1right)!!right)!!+9$$
              $$400 = 20^{-1+sqrt{9}}$$
              $$404 = 20+left(-1+9right)!!$$
              $$423 = left(left(left(2+0!right)!right)!!-1right)times9$$
              $$432 = left(left(2+0!right)!right)!!times1times9$$
              $$441 = left(left(left(2+0!right)!right)!!+1right)times9$$
              $$472 = 2^{-0!}timesleft(-1+9!!right)$$
              $$473 = 2^{-0!}timesleft(1+9!!right)$$
              $$480 = 20timesleft(1+sqrt{9}right)!$$
              $$504 = left(left(2+0!right)!right)!^{-1}times9!$$
              $$510 = -2+left(0!+1right)^{9}$$
              $$512 = 2^{0+1times9}$$
              $$514 = 2+left(0!+1right)^{9}$$
              $$519 = -201+left(left(sqrt{9}right)!right)!$$
              $$560 = frac{left(left(2+0!right)!+1right)!}{9}$$
              $$561 = -left(left(2+0!+1right)!!right)!!+9!!$$
              $$600 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!$$
              $$603 = 201timessqrt{9}$$
              $$615 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)!!$$
              $$630 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)!!$$
              $$671 = left(left(2+0!right)!right)!-left(1+left(left(sqrt{9}right)!right)!!right)$$
              $$672 = left(left(2+0!right)!right)!-1timesleft(left(sqrt{9}right)!right)!!$$
              $$673 = left(left(2+0!right)!right)!+1-left(left(sqrt{9}right)!right)!!$$
              $$696 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)!$$
              $$699 = -20-left(1-left(left(sqrt{9}right)!right)!right)$$
              $$700 = -20+1timesleft(left(sqrt{9}right)!right)!$$
              $$701 = left(left(2+0!right)!right)!-19$$
              $$705 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!!$$
              $$710 = left(left(2+0!right)!right)!-left(1+9right)$$
              $$711 = left(left(2+0!right)!right)!-1times9$$
              $$712 = left(left(2+0!right)!right)!+1-9$$
              $$713 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)$$
              $$714 = left(left(2+0!right)!right)!-1timesleft(sqrt{9}right)!$$
              $$715 = left(left(2+0!right)!right)!+1-left(sqrt{9}right)!$$
              $$716 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)$$
              $$717 = -2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
              $$718 = 2timesleft(0-1right)+left(left(sqrt{9}right)!right)!$$
              $$719 = 2times0-left(1-left(left(sqrt{9}right)!right)!right)$$
              $$720 = 2times0+1timesleft(left(sqrt{9}right)!right)!$$
              $$721 = 2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
              $$722 = 2+0+1timesleft(left(sqrt{9}right)!right)!$$
              $$723 = 2+0+1+left(left(sqrt{9}right)!right)!$$
              $$724 = 2+0!+1+left(left(sqrt{9}right)!right)!$$
              $$725 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
              $$726 = left(2+0!right)!+1timesleft(left(sqrt{9}right)!right)!$$
              $$727 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!$$
              $$728 = left(left(2+0!right)!right)!-left(1-9right)$$
              $$729 = left(2+0+1right)^{left(sqrt{9}right)!}$$
              $$730 = left(left(2+0!right)!right)!+1+9$$
              $$735 = left(left(2+0!right)!-1right)!!+left(left(sqrt{9}right)!right)!$$
              $$739 = 20-left(1-left(left(sqrt{9}right)!right)!right)$$
              $$740 = 20+1timesleft(left(sqrt{9}right)!right)!$$
              $$741 = 20+1+left(left(sqrt{9}right)!right)!$$
              $$744 = left(left(2+0!right)!right)!+left(1+sqrt{9}right)!$$
              $$766 = 2timesleft(-0!+left(-1+9right)!!right)$$
              $$767 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
              $$768 = 2timesleft(0+left(-1+9right)!!right)$$
              $$769 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!!$$
              $$770 = 2timesleft(0!+left(-1+9right)!!right)$$
              $$825 = -left(left(2+0!right)!-1right)!+9!!$$
              $$840 = frac{left(left(2+0!right)!+1right)!}{left(sqrt{9}right)!}$$
              $$896 = -left(left(2+0!right)!right)!!-left(1-9!!right)$$
              $$897 = -left(left(2+0!right)!right)!!+1times9!!$$
              $$898 = -left(left(2+0!right)!right)!!+1+9!!$$
              $$912 = left(left(2+0!right)!right)!!times19$$
              $$921 = 201+left(left(sqrt{9}right)!right)!$$
              $$924 = -20-left(1-9!!right)$$
              $$925 = -20+1times9!!$$
              $$926 = -20+1+9!!$$
              $$930 = -left(left(2+0!right)!-1right)!!+9!!$$
              $$937 = -left(2+0!+1right)!!+9!!$$
              $$938 = -left(2+0!right)!-left(1-9!!right)$$
              $$939 = -left(2+0!right)!+1times9!!$$
              $$940 = 20timesleft(-1+left(left(sqrt{9}right)!right)!!right)$$
              $$941 = -2-left(0!+1-9!!right)$$
              $$942 = -2-left(0+1-9!!right)$$
              $$943 = 2timesleft(0-1right)+9!!$$
              $$944 = 2times0-left(1-9!!right)$$
              $$945 = 2times0+1times9!!$$
              $$946 = 2-left(0+1-9!!right)$$
              $$947 = 2+0+1times9!!$$
              $$948 = 2+0+1+9!!$$
              $$949 = 2+0!+1+9!!$$
              $$950 = left(2+0!right)!-left(1-9!!right)$$
              $$951 = left(2+0!right)!+1times9!!$$
              $$952 = left(2+0!right)!+1+9!!$$
              $$953 = left(2+0!+1right)!!+9!!$$
              $$960 = 20times1timesleft(left(sqrt{9}right)!right)!!$$
              $$964 = 20-left(1-9!!right)$$
              $$965 = 20+1times9!!$$
              $$966 = 20+1+9!!$$
              $$969 = left(2+0!+1right)!+9!!$$
              $$980 = 20timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
              $$992 = left(left(2+0!right)!right)!!-left(1-9!!right)$$
              $$993 = left(left(2+0!right)!right)!!+1times9!!$$
              $$994 = left(left(2+0!right)!right)!!+1+9!!$$




              1001 through 10000:




              $$1008 = left(20+1right)timesleft(left(sqrt{9}right)!right)!!$$
              $$1024 = 2^{0+1+9}$$
              $$1050 = left(left(2+0!right)!+1right)!!+9!!$$
              $$1065 = left(left(2+0!right)!-1right)!+9!!$$
              $$1080 = left(left(2+0!right)!-1right)!times9$$
              $$1104 = left(left(2+0!right)!right)!+left(-1+9right)!!$$
              $$1146 = 201+9!!$$
              $$1152 = left(2+0!right)timesleft(-1+9right)!!$$
              $$1206 = 201timesleft(sqrt{9}right)!$$
              $$1296 = left(2+0!right)!^{1+sqrt{9}}$$
              $$1329 = left(left(2+0!+1right)!!right)!!+9!!$$
              $$1436 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!right)right)$$
              $$1438 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!right)right)$$
              $$1439 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
              $$1440 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!right)$$
              $$1441 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!$$
              $$1442 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!right)$$
              $$1444 = 2timesleft(0!+1+left(left(sqrt{9}right)!right)!right)$$
              $$1664 = left(left(2+0!right)!right)!-left(1-9!!right)$$
              $$1665 = left(left(2+0!right)!right)!+1times9!!$$
              $$1666 = left(left(2+0!right)!right)!+1+9!!$$
              $$1680 = frac{left(left(2+0!right)!+1right)!}{sqrt{9}}$$
              $$1809 = 201times9$$
              $$1886 = 2timesleft(-0!-left(1-9!!right)right)$$
              $$1888 = 2timesleft(0-left(1-9!!right)right)$$
              $$1890 = 2timesleft(0+1times9!!right)$$
              $$1892 = 2timesleft(0+1+9!!right)$$
              $$1894 = 2timesleft(0!+1+9!!right)$$
              $$1920 = 2^{-0!}timesleft(1+9right)!!$$
              $$2019 = 2019$$
              $$2048 = 2^{0!+1+9}$$
              $$2100 = 20timesleft(1+left(sqrt{9}right)!right)!!$$
              $$2145 = frac{left(left(left(2+0!right)!-1right)!!right)!!}{9!!}$$
              $$2157 = left(left(left(2+0!right)!right)!-1right)timessqrt{9}$$
              $$2160 = left(2+0+1right)timesleft(left(sqrt{9}right)!right)!$$
              $$2163 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!right)$$
              $$2187 = left(2+0!right)^{1+left(sqrt{9}right)!}$$
              $$2256 = left(left(left(2+0!right)!right)!!-1right)timesleft(left(sqrt{9}right)!right)!!$$
              $$2304 = left(2+0!right)!timesleft(-1+9right)!!$$
              $$2352 = left(left(2+0!right)!right)!!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
              $$2400 = 20timesleft(-1+left(sqrt{9}right)!right)!$$
              $$2520 = 2^{-0!}timesleft(1+left(sqrt{9}right)!right)!$$
              $$2832 = left(2+0!right)timesleft(-1+9!!right)$$
              $$2835 = left(2+0+1right)times9!!$$
              $$2838 = left(2+0!right)timesleft(1+9!!right)$$
              $$2880 = 2timesleft(0!+1right)timesleft(left(sqrt{9}right)!right)!$$
              $$3120 = -left(left(2+0!right)!right)!+left(1+9right)!!$$
              $$3375 = left(left(2+0!right)!-1right)!!^{sqrt{9}}$$
              $$3456 = left(left(2+0!+1right)!!right)!!times9$$
              $$3600 = left(left(2+0!right)!-1right)timesleft(left(sqrt{9}right)!right)!$$
              $$3780 = 2timesleft(0!+1right)times9!!$$
              $$3792 = -left(left(2+0!right)!right)!!+left(1+9right)!!$$
              $$3820 = -20+left(1+9right)!!$$
              $$3834 = -left(2+0!right)!+left(1+9right)!!$$
              $$3837 = -2-left(0!-left(1+9right)!!right)$$
              $$3838 = -2+0+left(1+9right)!!$$
              $$3839 = -left(2^{0}right)+left(1+9right)!!$$
              $$3840 = 2times0+left(1+9right)!!$$
              $$3841 = 2^{0}+left(1+9right)!!$$
              $$3842 = 2+0+left(1+9right)!!$$
              $$3843 = 2+0!+left(1+9right)!!$$
              $$3846 = left(2+0!right)!+left(1+9right)!!$$
              $$3860 = 20+left(1+9right)!!$$
              $$3888 = left(left(2+0!right)!right)!!+left(1+9right)!!$$
              $$4094 = -2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
              $$4095 = left(left(2+0!right)!+1right)!-9!!$$
              $$4096 = 2^{left(0!+1right)timesleft(sqrt{9}right)!}$$
              $$4098 = 2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
              $$4314 = left(left(left(2+0!right)!right)!-1right)timesleft(sqrt{9}right)!$$
              $$4320 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!$$
              $$4326 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!right)$$
              $$4480 = frac{left(left(2+0!+1right)!!right)!}{9}$$
              $$4560 = left(left(2+0!right)!right)!+left(1+9right)!!$$
              $$4725 = left(left(2+0!right)!-1right)times9!!$$
              $$4992 = left(left(2+0!right)!+1right)!-left(left(sqrt{9}right)!right)!!$$
              $$5020 = -20+left(1+left(sqrt{9}right)!right)!$$
              $$5031 = left(left(2+0!right)!+1right)!-9$$
              $$5034 = left(left(2+0!right)!+1right)!-left(sqrt{9}right)!$$
              $$5037 = left(left(2+0!right)!+1right)!-sqrt{9}$$
              $$5038 = -2+0+left(1+left(sqrt{9}right)!right)!$$
              $$5039 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!$$
              $$5040 = 2times0+left(1+left(sqrt{9}right)!right)!$$
              $$5041 = 2^{0}+left(1+left(sqrt{9}right)!right)!$$
              $$5042 = 2+0+left(1+left(sqrt{9}right)!right)!$$
              $$5043 = 2+0!+left(1+left(sqrt{9}right)!right)!$$
              $$5046 = left(2+0!right)!+left(1+left(sqrt{9}right)!right)!$$
              $$5049 = left(left(2+0!right)!+1right)!+9$$
              $$5060 = 20+left(1+left(sqrt{9}right)!right)!$$
              $$5088 = left(left(2+0!right)!right)!!+left(1+left(sqrt{9}right)!right)!$$
              $$5664 = left(2+0!right)!timesleft(-1+9!!right)$$
              $$5670 = left(2+0!right)!times1times9!!$$
              $$5676 = left(2+0!right)!timesleft(1+9!!right)$$
              $$5760 = left(left(2+0!right)!right)!timesleft(-1+9right)$$
              $$5985 = left(left(2+0!right)!+1right)!+9!!$$
              $$6471 = left(left(left(2+0!right)!right)!-1right)times9$$
              $$6480 = left(left(2+0!right)!right)!times1times9$$
              $$6489 = left(left(left(2+0!right)!right)!+1right)times9$$
              $$6561 = left(2+0!right)^{-1+9}$$
              $$6615 = left(left(2+0!right)!+1right)times9!!$$
              $$6720 = frac{left(left(2+0!+1right)!!right)!}{left(sqrt{9}right)!}$$
              $$6859 = left(20-1right)^{sqrt{9}}$$
              $$7200 = left(left(2+0!right)!right)!timesleft(1+9right)$$
              $$7560 = left(2+0!+1right)!!times9!!$$
              $$7678 = 2timesleft(-0!+left(1+9right)!!right)$$
              $$7680 = 20timesleft(-1+9right)!!$$
              $$7682 = 2timesleft(0!+left(1+9right)!!right)$$
              $$7776 = left(2+0!right)!^{-1+left(sqrt{9}right)!}$$
              $$8000 = 20^{1timessqrt{9}}$$
              $$8192 = 2timessqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
              $$9261 = left(20+1right)^{sqrt{9}}$$
              $$9648 = 201timesleft(left(sqrt{9}right)!right)!!$$






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                1














                I wrote a program to determine all representable numbers between 1 and 10000 following the rules, so this should be a comprehensive list.



                0 through 30:




                $$0 = 2times0timesleft(1+9right)$$
                $$1 = 20-19$$
                $$2 = 2+0timesleft(1+9right)$$
                $$3 = 2+0+1^{9}$$
                $$4 = 2-left(0+1-sqrt{9}right)$$
                $$5 = frac{20}{1+sqrt{9}}$$
                $$6 = -2-left(0+1-9right)$$
                $$7 = 2timesleft(0-1right)+9$$
                $$8 = 2times0-left(1-9right)$$
                $$9 = 2times0+1times9$$
                $$10 = 2-left(0+1-9right)$$
                $$11 = 2+0+1times9$$
                $$12 = 2+0+1+9$$
                $$13 = 20-left(1+left(sqrt{9}right)!right)$$
                $$14 = 20-1timesleft(sqrt{9}right)!$$
                $$15 = 20+1-left(sqrt{9}right)!$$
                $$16 = 2timesleft(0-left(1-9right)right)$$
                $$17 = -2+0+19$$
                $$18 = 2timesleft(0+1times9right)$$
                $$19 = 2times0+19$$
                $$20 = 2timesleft(0+1+9right)$$
                $$21 = 2+0+19$$
                $$22 = 20-left(1-sqrt{9}right)$$
                $$23 = 20+1timessqrt{9}$$
                $$24 = 20+1+sqrt{9}$$
                $$25 = 20-left(1-left(sqrt{9}right)!right)$$
                $$26 = 20+1timesleft(sqrt{9}right)!$$
                $$27 = left(2+0+1right)times9$$
                $$28 = 20-left(1-9right)$$
                $$29 = 20+1times9$$
                $$30 = 20+1+9$$




                31 through 100. Interestingly enough, 31 is the smallest number that cannot be done.




                $$32 = sqrt{2^{0+1+9}}$$
                $$33 = left(2+0!+1right)!+9$$
                $$35 = 20+left(-1+left(sqrt{9}right)!right)!!$$
                $$36 = 2timesleft(-0!+19right)$$
                $$38 = 2timesleft(0+19right)$$
                $$39 = 20+19$$
                $$40 = 20timesleft(-1+sqrt{9}right)$$
                $$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt{9}right)!right)$$
                $$42 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)$$
                $$43 = left(left(2+0!right)!right)!!+1-left(sqrt{9}right)!$$
                $$44 = 20+left(1+sqrt{9}right)!$$
                $$45 = left(left(2+0!right)!-1right)times9$$
                $$46 = 2timesleft(-0!+left(1+sqrt{9}right)!right)$$
                $$47 = 2times0-left(1-left(left(sqrt{9}right)!right)!!right)$$
                $$48 = 2timesleft(0+left(1+sqrt{9}right)!right)$$
                $$49 = 2-left(0+1-left(left(sqrt{9}right)!right)!!right)$$
                $$50 = 2timesleft(0!+left(1+sqrt{9}right)!right)$$
                $$51 = 2+0+1+left(left(sqrt{9}right)!right)!!$$
                $$52 = 2+0!+1+left(left(sqrt{9}right)!right)!!$$
                $$53 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                $$54 = left(2+0!right)!times1times9$$
                $$55 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!!$$
                $$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
                $$57 = left(20-1right)timessqrt{9}$$
                $$58 = left(left(2+0!right)!right)!!+1+9$$
                $$60 = 20times1timessqrt{9}$$
                $$62 = -2+left(0!+1right)^{left(sqrt{9}right)!}$$
                $$63 = left(20+1right)timessqrt{9}$$
                $$64 = 2^{0+1timesleft(sqrt{9}right)!}$$
                $$66 = 2+left(0!+1right)^{left(sqrt{9}right)!}$$
                $$67 = frac{201}{sqrt{9}}$$
                $$68 = 20+1timesleft(left(sqrt{9}right)!right)!!$$
                $$69 = -2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                $$71 = sqrt{2^{0}+left(1+left(sqrt{9}right)!right)!}$$
                $$72 = left(2+0!right)timesleft(1+sqrt{9}right)!$$
                $$73 = 2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                $$80 = 20timesleft(1+sqrt{9}right)$$
                $$81 = left(2+0!right)^{1+sqrt{9}}$$
                $$85 = -20+left(1+left(sqrt{9}right)!right)!!$$
                $$90 = frac{left(left(2+0!right)!right)!}{-1+9}$$
                $$92 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
                $$94 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
                $$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                $$96 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!!right)$$
                $$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt{9}right)!right)!!$$
                $$98 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!!right)$$
                $$99 = left(left(2+0!right)!+1right)!!-left(sqrt{9}right)!$$
                $$100 = 20timesleft(-1+left(sqrt{9}right)!right)$$




                101 through 1000:




                $$102 = left(left(2+0!right)!+1right)!!-sqrt{9}$$
                $$103 = -2+0+left(1+left(sqrt{9}right)!right)!!$$
                $$104 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!!$$
                $$105 = 2times0+left(1+left(sqrt{9}right)!right)!!$$
                $$106 = 2^{0}+left(1+left(sqrt{9}right)!right)!!$$
                $$107 = 2+0+left(1+left(sqrt{9}right)!right)!!$$
                $$108 = 2+0!+left(1+left(sqrt{9}right)!right)!!$$
                $$111 = left(left(2+0!right)!-1right)!-9$$
                $$114 = left(2+0!right)!times19$$
                $$117 = left(left(2+0!right)!-1right)!-sqrt{9}$$
                $$118 = -2+0+left(-1+left(sqrt{9}right)!right)!$$
                $$119 = -left(2^{0}right)+left(-1+left(sqrt{9}right)!right)!$$
                $$120 = 20times1timesleft(sqrt{9}right)!$$
                $$121 = 2^{0}+left(-1+left(sqrt{9}right)!right)!$$
                $$122 = 2+0+left(-1+left(sqrt{9}right)!right)!$$
                $$123 = left(left(2+0!right)!-1right)!+sqrt{9}$$
                $$125 = left(left(2+0!right)!-1right)^{sqrt{9}}$$
                $$126 = left(20+1right)timesleft(sqrt{9}right)!$$
                $$128 = 2^{0+1+left(sqrt{9}right)!}$$
                $$129 = left(left(2+0!right)!-1right)!+9$$
                $$135 = left(left(2+0!right)!-1right)!!times9$$
                $$140 = 20timesleft(1+left(sqrt{9}right)!right)$$
                $$141 = left(left(left(2+0!right)!right)!!-1right)timessqrt{9}$$
                $$142 = 2timessqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                $$144 = left(2+0!right)!timesleft(1+sqrt{9}right)!$$
                $$147 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                $$153 = 201-left(left(sqrt{9}right)!right)!!$$
                $$160 = 20timesleft(-1+9right)$$
                $$168 = left(left(2+0!right)!-1right)!+left(left(sqrt{9}right)!right)!!$$
                $$171 = left(20-1right)times9$$
                $$180 = 20times1times9$$
                $$189 = left(20+1right)times9$$
                $$192 = 201-9$$
                $$195 = 201-left(sqrt{9}right)!$$
                $$198 = 201-sqrt{9}$$
                $$200 = 20timesleft(1+9right)$$
                $$204 = 201+sqrt{9}$$
                $$207 = 201+left(sqrt{9}right)!$$
                $$208 = 2timesleft(-0!+left(1+left(sqrt{9}right)!right)!!right)$$
                $$210 = 201+9$$
                $$212 = 2timesleft(0!+left(1+left(sqrt{9}right)!right)!!right)$$
                $$216 = left(2+0!+1right)!times9$$
                $$224 = -left(left(2+0!right)!right)!-left(1-9!!right)$$
                $$225 = -left(left(2+0!right)!right)!+1times9!!$$
                $$226 = -left(left(2+0!right)!right)!+1+9!!$$
                $$238 = 2timesleft(-0!+left(-1+left(sqrt{9}right)!right)!right)$$
                $$240 = 2timesleft(0+left(-1+left(sqrt{9}right)!right)!right)$$
                $$242 = 2timesleft(0!+left(-1+left(sqrt{9}right)!right)!right)$$
                $$243 = left(2+0!right)^{-1+left(sqrt{9}right)!}$$
                $$249 = 201+left(left(sqrt{9}right)!right)!!$$
                $$256 = 2^{0-left(1-9right)}$$
                $$282 = left(left(left(2+0!right)!right)!!-1right)timesleft(sqrt{9}right)!$$
                $$288 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!!$$
                $$294 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                $$300 = 20timesleft(-1+left(sqrt{9}right)!right)!!$$
                $$315 = left(2+0!right)^{-1}times9!!$$
                $$336 = left(left(2+0!right)!right)!-left(-1+9right)!!$$
                $$343 = left(left(2+0!right)!+1right)^{sqrt{9}}$$
                $$360 = 2^{0-1}timesleft(left(sqrt{9}right)!right)!$$
                $$364 = -20+left(-1+9right)!!$$
                $$375 = left(left(2+0!+1right)!!right)!!-9$$
                $$378 = -left(2+0!right)!+left(-1+9right)!!$$
                $$380 = 20times19$$
                $$381 = -2-left(0!-left(-1+9right)!!right)$$
                $$382 = -2+0+left(-1+9right)!!$$
                $$383 = -left(2^{0}right)+left(-1+9right)!!$$
                $$384 = 2times0+left(-1+9right)!!$$
                $$385 = 2^{0}+left(-1+9right)!!$$
                $$386 = 2+0+left(-1+9right)!!$$
                $$387 = 2+0!+left(-1+9right)!!$$
                $$390 = left(2+0!right)!+left(-1+9right)!!$$
                $$393 = left(left(2+0!+1right)!!right)!!+9$$
                $$400 = 20^{-1+sqrt{9}}$$
                $$404 = 20+left(-1+9right)!!$$
                $$423 = left(left(left(2+0!right)!right)!!-1right)times9$$
                $$432 = left(left(2+0!right)!right)!!times1times9$$
                $$441 = left(left(left(2+0!right)!right)!!+1right)times9$$
                $$472 = 2^{-0!}timesleft(-1+9!!right)$$
                $$473 = 2^{-0!}timesleft(1+9!!right)$$
                $$480 = 20timesleft(1+sqrt{9}right)!$$
                $$504 = left(left(2+0!right)!right)!^{-1}times9!$$
                $$510 = -2+left(0!+1right)^{9}$$
                $$512 = 2^{0+1times9}$$
                $$514 = 2+left(0!+1right)^{9}$$
                $$519 = -201+left(left(sqrt{9}right)!right)!$$
                $$560 = frac{left(left(2+0!right)!+1right)!}{9}$$
                $$561 = -left(left(2+0!+1right)!!right)!!+9!!$$
                $$600 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!$$
                $$603 = 201timessqrt{9}$$
                $$615 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)!!$$
                $$630 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)!!$$
                $$671 = left(left(2+0!right)!right)!-left(1+left(left(sqrt{9}right)!right)!!right)$$
                $$672 = left(left(2+0!right)!right)!-1timesleft(left(sqrt{9}right)!right)!!$$
                $$673 = left(left(2+0!right)!right)!+1-left(left(sqrt{9}right)!right)!!$$
                $$696 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)!$$
                $$699 = -20-left(1-left(left(sqrt{9}right)!right)!right)$$
                $$700 = -20+1timesleft(left(sqrt{9}right)!right)!$$
                $$701 = left(left(2+0!right)!right)!-19$$
                $$705 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!!$$
                $$710 = left(left(2+0!right)!right)!-left(1+9right)$$
                $$711 = left(left(2+0!right)!right)!-1times9$$
                $$712 = left(left(2+0!right)!right)!+1-9$$
                $$713 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)$$
                $$714 = left(left(2+0!right)!right)!-1timesleft(sqrt{9}right)!$$
                $$715 = left(left(2+0!right)!right)!+1-left(sqrt{9}right)!$$
                $$716 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)$$
                $$717 = -2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
                $$718 = 2timesleft(0-1right)+left(left(sqrt{9}right)!right)!$$
                $$719 = 2times0-left(1-left(left(sqrt{9}right)!right)!right)$$
                $$720 = 2times0+1timesleft(left(sqrt{9}right)!right)!$$
                $$721 = 2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
                $$722 = 2+0+1timesleft(left(sqrt{9}right)!right)!$$
                $$723 = 2+0+1+left(left(sqrt{9}right)!right)!$$
                $$724 = 2+0!+1+left(left(sqrt{9}right)!right)!$$
                $$725 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
                $$726 = left(2+0!right)!+1timesleft(left(sqrt{9}right)!right)!$$
                $$727 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!$$
                $$728 = left(left(2+0!right)!right)!-left(1-9right)$$
                $$729 = left(2+0+1right)^{left(sqrt{9}right)!}$$
                $$730 = left(left(2+0!right)!right)!+1+9$$
                $$735 = left(left(2+0!right)!-1right)!!+left(left(sqrt{9}right)!right)!$$
                $$739 = 20-left(1-left(left(sqrt{9}right)!right)!right)$$
                $$740 = 20+1timesleft(left(sqrt{9}right)!right)!$$
                $$741 = 20+1+left(left(sqrt{9}right)!right)!$$
                $$744 = left(left(2+0!right)!right)!+left(1+sqrt{9}right)!$$
                $$766 = 2timesleft(-0!+left(-1+9right)!!right)$$
                $$767 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                $$768 = 2timesleft(0+left(-1+9right)!!right)$$
                $$769 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!!$$
                $$770 = 2timesleft(0!+left(-1+9right)!!right)$$
                $$825 = -left(left(2+0!right)!-1right)!+9!!$$
                $$840 = frac{left(left(2+0!right)!+1right)!}{left(sqrt{9}right)!}$$
                $$896 = -left(left(2+0!right)!right)!!-left(1-9!!right)$$
                $$897 = -left(left(2+0!right)!right)!!+1times9!!$$
                $$898 = -left(left(2+0!right)!right)!!+1+9!!$$
                $$912 = left(left(2+0!right)!right)!!times19$$
                $$921 = 201+left(left(sqrt{9}right)!right)!$$
                $$924 = -20-left(1-9!!right)$$
                $$925 = -20+1times9!!$$
                $$926 = -20+1+9!!$$
                $$930 = -left(left(2+0!right)!-1right)!!+9!!$$
                $$937 = -left(2+0!+1right)!!+9!!$$
                $$938 = -left(2+0!right)!-left(1-9!!right)$$
                $$939 = -left(2+0!right)!+1times9!!$$
                $$940 = 20timesleft(-1+left(left(sqrt{9}right)!right)!!right)$$
                $$941 = -2-left(0!+1-9!!right)$$
                $$942 = -2-left(0+1-9!!right)$$
                $$943 = 2timesleft(0-1right)+9!!$$
                $$944 = 2times0-left(1-9!!right)$$
                $$945 = 2times0+1times9!!$$
                $$946 = 2-left(0+1-9!!right)$$
                $$947 = 2+0+1times9!!$$
                $$948 = 2+0+1+9!!$$
                $$949 = 2+0!+1+9!!$$
                $$950 = left(2+0!right)!-left(1-9!!right)$$
                $$951 = left(2+0!right)!+1times9!!$$
                $$952 = left(2+0!right)!+1+9!!$$
                $$953 = left(2+0!+1right)!!+9!!$$
                $$960 = 20times1timesleft(left(sqrt{9}right)!right)!!$$
                $$964 = 20-left(1-9!!right)$$
                $$965 = 20+1times9!!$$
                $$966 = 20+1+9!!$$
                $$969 = left(2+0!+1right)!+9!!$$
                $$980 = 20timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                $$992 = left(left(2+0!right)!right)!!-left(1-9!!right)$$
                $$993 = left(left(2+0!right)!right)!!+1times9!!$$
                $$994 = left(left(2+0!right)!right)!!+1+9!!$$




                1001 through 10000:




                $$1008 = left(20+1right)timesleft(left(sqrt{9}right)!right)!!$$
                $$1024 = 2^{0+1+9}$$
                $$1050 = left(left(2+0!right)!+1right)!!+9!!$$
                $$1065 = left(left(2+0!right)!-1right)!+9!!$$
                $$1080 = left(left(2+0!right)!-1right)!times9$$
                $$1104 = left(left(2+0!right)!right)!+left(-1+9right)!!$$
                $$1146 = 201+9!!$$
                $$1152 = left(2+0!right)timesleft(-1+9right)!!$$
                $$1206 = 201timesleft(sqrt{9}right)!$$
                $$1296 = left(2+0!right)!^{1+sqrt{9}}$$
                $$1329 = left(left(2+0!+1right)!!right)!!+9!!$$
                $$1436 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!right)right)$$
                $$1438 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!right)right)$$
                $$1439 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
                $$1440 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!right)$$
                $$1441 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!$$
                $$1442 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!right)$$
                $$1444 = 2timesleft(0!+1+left(left(sqrt{9}right)!right)!right)$$
                $$1664 = left(left(2+0!right)!right)!-left(1-9!!right)$$
                $$1665 = left(left(2+0!right)!right)!+1times9!!$$
                $$1666 = left(left(2+0!right)!right)!+1+9!!$$
                $$1680 = frac{left(left(2+0!right)!+1right)!}{sqrt{9}}$$
                $$1809 = 201times9$$
                $$1886 = 2timesleft(-0!-left(1-9!!right)right)$$
                $$1888 = 2timesleft(0-left(1-9!!right)right)$$
                $$1890 = 2timesleft(0+1times9!!right)$$
                $$1892 = 2timesleft(0+1+9!!right)$$
                $$1894 = 2timesleft(0!+1+9!!right)$$
                $$1920 = 2^{-0!}timesleft(1+9right)!!$$
                $$2019 = 2019$$
                $$2048 = 2^{0!+1+9}$$
                $$2100 = 20timesleft(1+left(sqrt{9}right)!right)!!$$
                $$2145 = frac{left(left(left(2+0!right)!-1right)!!right)!!}{9!!}$$
                $$2157 = left(left(left(2+0!right)!right)!-1right)timessqrt{9}$$
                $$2160 = left(2+0+1right)timesleft(left(sqrt{9}right)!right)!$$
                $$2163 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!right)$$
                $$2187 = left(2+0!right)^{1+left(sqrt{9}right)!}$$
                $$2256 = left(left(left(2+0!right)!right)!!-1right)timesleft(left(sqrt{9}right)!right)!!$$
                $$2304 = left(2+0!right)!timesleft(-1+9right)!!$$
                $$2352 = left(left(2+0!right)!right)!!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                $$2400 = 20timesleft(-1+left(sqrt{9}right)!right)!$$
                $$2520 = 2^{-0!}timesleft(1+left(sqrt{9}right)!right)!$$
                $$2832 = left(2+0!right)timesleft(-1+9!!right)$$
                $$2835 = left(2+0+1right)times9!!$$
                $$2838 = left(2+0!right)timesleft(1+9!!right)$$
                $$2880 = 2timesleft(0!+1right)timesleft(left(sqrt{9}right)!right)!$$
                $$3120 = -left(left(2+0!right)!right)!+left(1+9right)!!$$
                $$3375 = left(left(2+0!right)!-1right)!!^{sqrt{9}}$$
                $$3456 = left(left(2+0!+1right)!!right)!!times9$$
                $$3600 = left(left(2+0!right)!-1right)timesleft(left(sqrt{9}right)!right)!$$
                $$3780 = 2timesleft(0!+1right)times9!!$$
                $$3792 = -left(left(2+0!right)!right)!!+left(1+9right)!!$$
                $$3820 = -20+left(1+9right)!!$$
                $$3834 = -left(2+0!right)!+left(1+9right)!!$$
                $$3837 = -2-left(0!-left(1+9right)!!right)$$
                $$3838 = -2+0+left(1+9right)!!$$
                $$3839 = -left(2^{0}right)+left(1+9right)!!$$
                $$3840 = 2times0+left(1+9right)!!$$
                $$3841 = 2^{0}+left(1+9right)!!$$
                $$3842 = 2+0+left(1+9right)!!$$
                $$3843 = 2+0!+left(1+9right)!!$$
                $$3846 = left(2+0!right)!+left(1+9right)!!$$
                $$3860 = 20+left(1+9right)!!$$
                $$3888 = left(left(2+0!right)!right)!!+left(1+9right)!!$$
                $$4094 = -2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                $$4095 = left(left(2+0!right)!+1right)!-9!!$$
                $$4096 = 2^{left(0!+1right)timesleft(sqrt{9}right)!}$$
                $$4098 = 2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                $$4314 = left(left(left(2+0!right)!right)!-1right)timesleft(sqrt{9}right)!$$
                $$4320 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!$$
                $$4326 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!right)$$
                $$4480 = frac{left(left(2+0!+1right)!!right)!}{9}$$
                $$4560 = left(left(2+0!right)!right)!+left(1+9right)!!$$
                $$4725 = left(left(2+0!right)!-1right)times9!!$$
                $$4992 = left(left(2+0!right)!+1right)!-left(left(sqrt{9}right)!right)!!$$
                $$5020 = -20+left(1+left(sqrt{9}right)!right)!$$
                $$5031 = left(left(2+0!right)!+1right)!-9$$
                $$5034 = left(left(2+0!right)!+1right)!-left(sqrt{9}right)!$$
                $$5037 = left(left(2+0!right)!+1right)!-sqrt{9}$$
                $$5038 = -2+0+left(1+left(sqrt{9}right)!right)!$$
                $$5039 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!$$
                $$5040 = 2times0+left(1+left(sqrt{9}right)!right)!$$
                $$5041 = 2^{0}+left(1+left(sqrt{9}right)!right)!$$
                $$5042 = 2+0+left(1+left(sqrt{9}right)!right)!$$
                $$5043 = 2+0!+left(1+left(sqrt{9}right)!right)!$$
                $$5046 = left(2+0!right)!+left(1+left(sqrt{9}right)!right)!$$
                $$5049 = left(left(2+0!right)!+1right)!+9$$
                $$5060 = 20+left(1+left(sqrt{9}right)!right)!$$
                $$5088 = left(left(2+0!right)!right)!!+left(1+left(sqrt{9}right)!right)!$$
                $$5664 = left(2+0!right)!timesleft(-1+9!!right)$$
                $$5670 = left(2+0!right)!times1times9!!$$
                $$5676 = left(2+0!right)!timesleft(1+9!!right)$$
                $$5760 = left(left(2+0!right)!right)!timesleft(-1+9right)$$
                $$5985 = left(left(2+0!right)!+1right)!+9!!$$
                $$6471 = left(left(left(2+0!right)!right)!-1right)times9$$
                $$6480 = left(left(2+0!right)!right)!times1times9$$
                $$6489 = left(left(left(2+0!right)!right)!+1right)times9$$
                $$6561 = left(2+0!right)^{-1+9}$$
                $$6615 = left(left(2+0!right)!+1right)times9!!$$
                $$6720 = frac{left(left(2+0!+1right)!!right)!}{left(sqrt{9}right)!}$$
                $$6859 = left(20-1right)^{sqrt{9}}$$
                $$7200 = left(left(2+0!right)!right)!timesleft(1+9right)$$
                $$7560 = left(2+0!+1right)!!times9!!$$
                $$7678 = 2timesleft(-0!+left(1+9right)!!right)$$
                $$7680 = 20timesleft(-1+9right)!!$$
                $$7682 = 2timesleft(0!+left(1+9right)!!right)$$
                $$7776 = left(2+0!right)!^{-1+left(sqrt{9}right)!}$$
                $$8000 = 20^{1timessqrt{9}}$$
                $$8192 = 2timessqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                $$9261 = left(20+1right)^{sqrt{9}}$$
                $$9648 = 201timesleft(left(sqrt{9}right)!right)!!$$






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                  I wrote a program to determine all representable numbers between 1 and 10000 following the rules, so this should be a comprehensive list.



                  0 through 30:




                  $$0 = 2times0timesleft(1+9right)$$
                  $$1 = 20-19$$
                  $$2 = 2+0timesleft(1+9right)$$
                  $$3 = 2+0+1^{9}$$
                  $$4 = 2-left(0+1-sqrt{9}right)$$
                  $$5 = frac{20}{1+sqrt{9}}$$
                  $$6 = -2-left(0+1-9right)$$
                  $$7 = 2timesleft(0-1right)+9$$
                  $$8 = 2times0-left(1-9right)$$
                  $$9 = 2times0+1times9$$
                  $$10 = 2-left(0+1-9right)$$
                  $$11 = 2+0+1times9$$
                  $$12 = 2+0+1+9$$
                  $$13 = 20-left(1+left(sqrt{9}right)!right)$$
                  $$14 = 20-1timesleft(sqrt{9}right)!$$
                  $$15 = 20+1-left(sqrt{9}right)!$$
                  $$16 = 2timesleft(0-left(1-9right)right)$$
                  $$17 = -2+0+19$$
                  $$18 = 2timesleft(0+1times9right)$$
                  $$19 = 2times0+19$$
                  $$20 = 2timesleft(0+1+9right)$$
                  $$21 = 2+0+19$$
                  $$22 = 20-left(1-sqrt{9}right)$$
                  $$23 = 20+1timessqrt{9}$$
                  $$24 = 20+1+sqrt{9}$$
                  $$25 = 20-left(1-left(sqrt{9}right)!right)$$
                  $$26 = 20+1timesleft(sqrt{9}right)!$$
                  $$27 = left(2+0+1right)times9$$
                  $$28 = 20-left(1-9right)$$
                  $$29 = 20+1times9$$
                  $$30 = 20+1+9$$




                  31 through 100. Interestingly enough, 31 is the smallest number that cannot be done.




                  $$32 = sqrt{2^{0+1+9}}$$
                  $$33 = left(2+0!+1right)!+9$$
                  $$35 = 20+left(-1+left(sqrt{9}right)!right)!!$$
                  $$36 = 2timesleft(-0!+19right)$$
                  $$38 = 2timesleft(0+19right)$$
                  $$39 = 20+19$$
                  $$40 = 20timesleft(-1+sqrt{9}right)$$
                  $$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt{9}right)!right)$$
                  $$42 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)$$
                  $$43 = left(left(2+0!right)!right)!!+1-left(sqrt{9}right)!$$
                  $$44 = 20+left(1+sqrt{9}right)!$$
                  $$45 = left(left(2+0!right)!-1right)times9$$
                  $$46 = 2timesleft(-0!+left(1+sqrt{9}right)!right)$$
                  $$47 = 2times0-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$48 = 2timesleft(0+left(1+sqrt{9}right)!right)$$
                  $$49 = 2-left(0+1-left(left(sqrt{9}right)!right)!!right)$$
                  $$50 = 2timesleft(0!+left(1+sqrt{9}right)!right)$$
                  $$51 = 2+0+1+left(left(sqrt{9}right)!right)!!$$
                  $$52 = 2+0!+1+left(left(sqrt{9}right)!right)!!$$
                  $$53 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$54 = left(2+0!right)!times1times9$$
                  $$55 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!!$$
                  $$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
                  $$57 = left(20-1right)timessqrt{9}$$
                  $$58 = left(left(2+0!right)!right)!!+1+9$$
                  $$60 = 20times1timessqrt{9}$$
                  $$62 = -2+left(0!+1right)^{left(sqrt{9}right)!}$$
                  $$63 = left(20+1right)timessqrt{9}$$
                  $$64 = 2^{0+1timesleft(sqrt{9}right)!}$$
                  $$66 = 2+left(0!+1right)^{left(sqrt{9}right)!}$$
                  $$67 = frac{201}{sqrt{9}}$$
                  $$68 = 20+1timesleft(left(sqrt{9}right)!right)!!$$
                  $$69 = -2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                  $$71 = sqrt{2^{0}+left(1+left(sqrt{9}right)!right)!}$$
                  $$72 = left(2+0!right)timesleft(1+sqrt{9}right)!$$
                  $$73 = 2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                  $$80 = 20timesleft(1+sqrt{9}right)$$
                  $$81 = left(2+0!right)^{1+sqrt{9}}$$
                  $$85 = -20+left(1+left(sqrt{9}right)!right)!!$$
                  $$90 = frac{left(left(2+0!right)!right)!}{-1+9}$$
                  $$92 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
                  $$94 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
                  $$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$96 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!!right)$$
                  $$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt{9}right)!right)!!$$
                  $$98 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!!right)$$
                  $$99 = left(left(2+0!right)!+1right)!!-left(sqrt{9}right)!$$
                  $$100 = 20timesleft(-1+left(sqrt{9}right)!right)$$




                  101 through 1000:




                  $$102 = left(left(2+0!right)!+1right)!!-sqrt{9}$$
                  $$103 = -2+0+left(1+left(sqrt{9}right)!right)!!$$
                  $$104 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!!$$
                  $$105 = 2times0+left(1+left(sqrt{9}right)!right)!!$$
                  $$106 = 2^{0}+left(1+left(sqrt{9}right)!right)!!$$
                  $$107 = 2+0+left(1+left(sqrt{9}right)!right)!!$$
                  $$108 = 2+0!+left(1+left(sqrt{9}right)!right)!!$$
                  $$111 = left(left(2+0!right)!-1right)!-9$$
                  $$114 = left(2+0!right)!times19$$
                  $$117 = left(left(2+0!right)!-1right)!-sqrt{9}$$
                  $$118 = -2+0+left(-1+left(sqrt{9}right)!right)!$$
                  $$119 = -left(2^{0}right)+left(-1+left(sqrt{9}right)!right)!$$
                  $$120 = 20times1timesleft(sqrt{9}right)!$$
                  $$121 = 2^{0}+left(-1+left(sqrt{9}right)!right)!$$
                  $$122 = 2+0+left(-1+left(sqrt{9}right)!right)!$$
                  $$123 = left(left(2+0!right)!-1right)!+sqrt{9}$$
                  $$125 = left(left(2+0!right)!-1right)^{sqrt{9}}$$
                  $$126 = left(20+1right)timesleft(sqrt{9}right)!$$
                  $$128 = 2^{0+1+left(sqrt{9}right)!}$$
                  $$129 = left(left(2+0!right)!-1right)!+9$$
                  $$135 = left(left(2+0!right)!-1right)!!times9$$
                  $$140 = 20timesleft(1+left(sqrt{9}right)!right)$$
                  $$141 = left(left(left(2+0!right)!right)!!-1right)timessqrt{9}$$
                  $$142 = 2timessqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                  $$144 = left(2+0!right)!timesleft(1+sqrt{9}right)!$$
                  $$147 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$153 = 201-left(left(sqrt{9}right)!right)!!$$
                  $$160 = 20timesleft(-1+9right)$$
                  $$168 = left(left(2+0!right)!-1right)!+left(left(sqrt{9}right)!right)!!$$
                  $$171 = left(20-1right)times9$$
                  $$180 = 20times1times9$$
                  $$189 = left(20+1right)times9$$
                  $$192 = 201-9$$
                  $$195 = 201-left(sqrt{9}right)!$$
                  $$198 = 201-sqrt{9}$$
                  $$200 = 20timesleft(1+9right)$$
                  $$204 = 201+sqrt{9}$$
                  $$207 = 201+left(sqrt{9}right)!$$
                  $$208 = 2timesleft(-0!+left(1+left(sqrt{9}right)!right)!!right)$$
                  $$210 = 201+9$$
                  $$212 = 2timesleft(0!+left(1+left(sqrt{9}right)!right)!!right)$$
                  $$216 = left(2+0!+1right)!times9$$
                  $$224 = -left(left(2+0!right)!right)!-left(1-9!!right)$$
                  $$225 = -left(left(2+0!right)!right)!+1times9!!$$
                  $$226 = -left(left(2+0!right)!right)!+1+9!!$$
                  $$238 = 2timesleft(-0!+left(-1+left(sqrt{9}right)!right)!right)$$
                  $$240 = 2timesleft(0+left(-1+left(sqrt{9}right)!right)!right)$$
                  $$242 = 2timesleft(0!+left(-1+left(sqrt{9}right)!right)!right)$$
                  $$243 = left(2+0!right)^{-1+left(sqrt{9}right)!}$$
                  $$249 = 201+left(left(sqrt{9}right)!right)!!$$
                  $$256 = 2^{0-left(1-9right)}$$
                  $$282 = left(left(left(2+0!right)!right)!!-1right)timesleft(sqrt{9}right)!$$
                  $$288 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!!$$
                  $$294 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$300 = 20timesleft(-1+left(sqrt{9}right)!right)!!$$
                  $$315 = left(2+0!right)^{-1}times9!!$$
                  $$336 = left(left(2+0!right)!right)!-left(-1+9right)!!$$
                  $$343 = left(left(2+0!right)!+1right)^{sqrt{9}}$$
                  $$360 = 2^{0-1}timesleft(left(sqrt{9}right)!right)!$$
                  $$364 = -20+left(-1+9right)!!$$
                  $$375 = left(left(2+0!+1right)!!right)!!-9$$
                  $$378 = -left(2+0!right)!+left(-1+9right)!!$$
                  $$380 = 20times19$$
                  $$381 = -2-left(0!-left(-1+9right)!!right)$$
                  $$382 = -2+0+left(-1+9right)!!$$
                  $$383 = -left(2^{0}right)+left(-1+9right)!!$$
                  $$384 = 2times0+left(-1+9right)!!$$
                  $$385 = 2^{0}+left(-1+9right)!!$$
                  $$386 = 2+0+left(-1+9right)!!$$
                  $$387 = 2+0!+left(-1+9right)!!$$
                  $$390 = left(2+0!right)!+left(-1+9right)!!$$
                  $$393 = left(left(2+0!+1right)!!right)!!+9$$
                  $$400 = 20^{-1+sqrt{9}}$$
                  $$404 = 20+left(-1+9right)!!$$
                  $$423 = left(left(left(2+0!right)!right)!!-1right)times9$$
                  $$432 = left(left(2+0!right)!right)!!times1times9$$
                  $$441 = left(left(left(2+0!right)!right)!!+1right)times9$$
                  $$472 = 2^{-0!}timesleft(-1+9!!right)$$
                  $$473 = 2^{-0!}timesleft(1+9!!right)$$
                  $$480 = 20timesleft(1+sqrt{9}right)!$$
                  $$504 = left(left(2+0!right)!right)!^{-1}times9!$$
                  $$510 = -2+left(0!+1right)^{9}$$
                  $$512 = 2^{0+1times9}$$
                  $$514 = 2+left(0!+1right)^{9}$$
                  $$519 = -201+left(left(sqrt{9}right)!right)!$$
                  $$560 = frac{left(left(2+0!right)!+1right)!}{9}$$
                  $$561 = -left(left(2+0!+1right)!!right)!!+9!!$$
                  $$600 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!$$
                  $$603 = 201timessqrt{9}$$
                  $$615 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)!!$$
                  $$630 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)!!$$
                  $$671 = left(left(2+0!right)!right)!-left(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$672 = left(left(2+0!right)!right)!-1timesleft(left(sqrt{9}right)!right)!!$$
                  $$673 = left(left(2+0!right)!right)!+1-left(left(sqrt{9}right)!right)!!$$
                  $$696 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)!$$
                  $$699 = -20-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$700 = -20+1timesleft(left(sqrt{9}right)!right)!$$
                  $$701 = left(left(2+0!right)!right)!-19$$
                  $$705 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!!$$
                  $$710 = left(left(2+0!right)!right)!-left(1+9right)$$
                  $$711 = left(left(2+0!right)!right)!-1times9$$
                  $$712 = left(left(2+0!right)!right)!+1-9$$
                  $$713 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)$$
                  $$714 = left(left(2+0!right)!right)!-1timesleft(sqrt{9}right)!$$
                  $$715 = left(left(2+0!right)!right)!+1-left(sqrt{9}right)!$$
                  $$716 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)$$
                  $$717 = -2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
                  $$718 = 2timesleft(0-1right)+left(left(sqrt{9}right)!right)!$$
                  $$719 = 2times0-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$720 = 2times0+1timesleft(left(sqrt{9}right)!right)!$$
                  $$721 = 2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
                  $$722 = 2+0+1timesleft(left(sqrt{9}right)!right)!$$
                  $$723 = 2+0+1+left(left(sqrt{9}right)!right)!$$
                  $$724 = 2+0!+1+left(left(sqrt{9}right)!right)!$$
                  $$725 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$726 = left(2+0!right)!+1timesleft(left(sqrt{9}right)!right)!$$
                  $$727 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!$$
                  $$728 = left(left(2+0!right)!right)!-left(1-9right)$$
                  $$729 = left(2+0+1right)^{left(sqrt{9}right)!}$$
                  $$730 = left(left(2+0!right)!right)!+1+9$$
                  $$735 = left(left(2+0!right)!-1right)!!+left(left(sqrt{9}right)!right)!$$
                  $$739 = 20-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$740 = 20+1timesleft(left(sqrt{9}right)!right)!$$
                  $$741 = 20+1+left(left(sqrt{9}right)!right)!$$
                  $$744 = left(left(2+0!right)!right)!+left(1+sqrt{9}right)!$$
                  $$766 = 2timesleft(-0!+left(-1+9right)!!right)$$
                  $$767 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$768 = 2timesleft(0+left(-1+9right)!!right)$$
                  $$769 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!!$$
                  $$770 = 2timesleft(0!+left(-1+9right)!!right)$$
                  $$825 = -left(left(2+0!right)!-1right)!+9!!$$
                  $$840 = frac{left(left(2+0!right)!+1right)!}{left(sqrt{9}right)!}$$
                  $$896 = -left(left(2+0!right)!right)!!-left(1-9!!right)$$
                  $$897 = -left(left(2+0!right)!right)!!+1times9!!$$
                  $$898 = -left(left(2+0!right)!right)!!+1+9!!$$
                  $$912 = left(left(2+0!right)!right)!!times19$$
                  $$921 = 201+left(left(sqrt{9}right)!right)!$$
                  $$924 = -20-left(1-9!!right)$$
                  $$925 = -20+1times9!!$$
                  $$926 = -20+1+9!!$$
                  $$930 = -left(left(2+0!right)!-1right)!!+9!!$$
                  $$937 = -left(2+0!+1right)!!+9!!$$
                  $$938 = -left(2+0!right)!-left(1-9!!right)$$
                  $$939 = -left(2+0!right)!+1times9!!$$
                  $$940 = 20timesleft(-1+left(left(sqrt{9}right)!right)!!right)$$
                  $$941 = -2-left(0!+1-9!!right)$$
                  $$942 = -2-left(0+1-9!!right)$$
                  $$943 = 2timesleft(0-1right)+9!!$$
                  $$944 = 2times0-left(1-9!!right)$$
                  $$945 = 2times0+1times9!!$$
                  $$946 = 2-left(0+1-9!!right)$$
                  $$947 = 2+0+1times9!!$$
                  $$948 = 2+0+1+9!!$$
                  $$949 = 2+0!+1+9!!$$
                  $$950 = left(2+0!right)!-left(1-9!!right)$$
                  $$951 = left(2+0!right)!+1times9!!$$
                  $$952 = left(2+0!right)!+1+9!!$$
                  $$953 = left(2+0!+1right)!!+9!!$$
                  $$960 = 20times1timesleft(left(sqrt{9}right)!right)!!$$
                  $$964 = 20-left(1-9!!right)$$
                  $$965 = 20+1times9!!$$
                  $$966 = 20+1+9!!$$
                  $$969 = left(2+0!+1right)!+9!!$$
                  $$980 = 20timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$992 = left(left(2+0!right)!right)!!-left(1-9!!right)$$
                  $$993 = left(left(2+0!right)!right)!!+1times9!!$$
                  $$994 = left(left(2+0!right)!right)!!+1+9!!$$




                  1001 through 10000:




                  $$1008 = left(20+1right)timesleft(left(sqrt{9}right)!right)!!$$
                  $$1024 = 2^{0+1+9}$$
                  $$1050 = left(left(2+0!right)!+1right)!!+9!!$$
                  $$1065 = left(left(2+0!right)!-1right)!+9!!$$
                  $$1080 = left(left(2+0!right)!-1right)!times9$$
                  $$1104 = left(left(2+0!right)!right)!+left(-1+9right)!!$$
                  $$1146 = 201+9!!$$
                  $$1152 = left(2+0!right)timesleft(-1+9right)!!$$
                  $$1206 = 201timesleft(sqrt{9}right)!$$
                  $$1296 = left(2+0!right)!^{1+sqrt{9}}$$
                  $$1329 = left(left(2+0!+1right)!!right)!!+9!!$$
                  $$1436 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!right)right)$$
                  $$1438 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!right)right)$$
                  $$1439 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$1440 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!right)$$
                  $$1441 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!$$
                  $$1442 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!right)$$
                  $$1444 = 2timesleft(0!+1+left(left(sqrt{9}right)!right)!right)$$
                  $$1664 = left(left(2+0!right)!right)!-left(1-9!!right)$$
                  $$1665 = left(left(2+0!right)!right)!+1times9!!$$
                  $$1666 = left(left(2+0!right)!right)!+1+9!!$$
                  $$1680 = frac{left(left(2+0!right)!+1right)!}{sqrt{9}}$$
                  $$1809 = 201times9$$
                  $$1886 = 2timesleft(-0!-left(1-9!!right)right)$$
                  $$1888 = 2timesleft(0-left(1-9!!right)right)$$
                  $$1890 = 2timesleft(0+1times9!!right)$$
                  $$1892 = 2timesleft(0+1+9!!right)$$
                  $$1894 = 2timesleft(0!+1+9!!right)$$
                  $$1920 = 2^{-0!}timesleft(1+9right)!!$$
                  $$2019 = 2019$$
                  $$2048 = 2^{0!+1+9}$$
                  $$2100 = 20timesleft(1+left(sqrt{9}right)!right)!!$$
                  $$2145 = frac{left(left(left(2+0!right)!-1right)!!right)!!}{9!!}$$
                  $$2157 = left(left(left(2+0!right)!right)!-1right)timessqrt{9}$$
                  $$2160 = left(2+0+1right)timesleft(left(sqrt{9}right)!right)!$$
                  $$2163 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!right)$$
                  $$2187 = left(2+0!right)^{1+left(sqrt{9}right)!}$$
                  $$2256 = left(left(left(2+0!right)!right)!!-1right)timesleft(left(sqrt{9}right)!right)!!$$
                  $$2304 = left(2+0!right)!timesleft(-1+9right)!!$$
                  $$2352 = left(left(2+0!right)!right)!!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$2400 = 20timesleft(-1+left(sqrt{9}right)!right)!$$
                  $$2520 = 2^{-0!}timesleft(1+left(sqrt{9}right)!right)!$$
                  $$2832 = left(2+0!right)timesleft(-1+9!!right)$$
                  $$2835 = left(2+0+1right)times9!!$$
                  $$2838 = left(2+0!right)timesleft(1+9!!right)$$
                  $$2880 = 2timesleft(0!+1right)timesleft(left(sqrt{9}right)!right)!$$
                  $$3120 = -left(left(2+0!right)!right)!+left(1+9right)!!$$
                  $$3375 = left(left(2+0!right)!-1right)!!^{sqrt{9}}$$
                  $$3456 = left(left(2+0!+1right)!!right)!!times9$$
                  $$3600 = left(left(2+0!right)!-1right)timesleft(left(sqrt{9}right)!right)!$$
                  $$3780 = 2timesleft(0!+1right)times9!!$$
                  $$3792 = -left(left(2+0!right)!right)!!+left(1+9right)!!$$
                  $$3820 = -20+left(1+9right)!!$$
                  $$3834 = -left(2+0!right)!+left(1+9right)!!$$
                  $$3837 = -2-left(0!-left(1+9right)!!right)$$
                  $$3838 = -2+0+left(1+9right)!!$$
                  $$3839 = -left(2^{0}right)+left(1+9right)!!$$
                  $$3840 = 2times0+left(1+9right)!!$$
                  $$3841 = 2^{0}+left(1+9right)!!$$
                  $$3842 = 2+0+left(1+9right)!!$$
                  $$3843 = 2+0!+left(1+9right)!!$$
                  $$3846 = left(2+0!right)!+left(1+9right)!!$$
                  $$3860 = 20+left(1+9right)!!$$
                  $$3888 = left(left(2+0!right)!right)!!+left(1+9right)!!$$
                  $$4094 = -2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                  $$4095 = left(left(2+0!right)!+1right)!-9!!$$
                  $$4096 = 2^{left(0!+1right)timesleft(sqrt{9}right)!}$$
                  $$4098 = 2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                  $$4314 = left(left(left(2+0!right)!right)!-1right)timesleft(sqrt{9}right)!$$
                  $$4320 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!$$
                  $$4326 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!right)$$
                  $$4480 = frac{left(left(2+0!+1right)!!right)!}{9}$$
                  $$4560 = left(left(2+0!right)!right)!+left(1+9right)!!$$
                  $$4725 = left(left(2+0!right)!-1right)times9!!$$
                  $$4992 = left(left(2+0!right)!+1right)!-left(left(sqrt{9}right)!right)!!$$
                  $$5020 = -20+left(1+left(sqrt{9}right)!right)!$$
                  $$5031 = left(left(2+0!right)!+1right)!-9$$
                  $$5034 = left(left(2+0!right)!+1right)!-left(sqrt{9}right)!$$
                  $$5037 = left(left(2+0!right)!+1right)!-sqrt{9}$$
                  $$5038 = -2+0+left(1+left(sqrt{9}right)!right)!$$
                  $$5039 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!$$
                  $$5040 = 2times0+left(1+left(sqrt{9}right)!right)!$$
                  $$5041 = 2^{0}+left(1+left(sqrt{9}right)!right)!$$
                  $$5042 = 2+0+left(1+left(sqrt{9}right)!right)!$$
                  $$5043 = 2+0!+left(1+left(sqrt{9}right)!right)!$$
                  $$5046 = left(2+0!right)!+left(1+left(sqrt{9}right)!right)!$$
                  $$5049 = left(left(2+0!right)!+1right)!+9$$
                  $$5060 = 20+left(1+left(sqrt{9}right)!right)!$$
                  $$5088 = left(left(2+0!right)!right)!!+left(1+left(sqrt{9}right)!right)!$$
                  $$5664 = left(2+0!right)!timesleft(-1+9!!right)$$
                  $$5670 = left(2+0!right)!times1times9!!$$
                  $$5676 = left(2+0!right)!timesleft(1+9!!right)$$
                  $$5760 = left(left(2+0!right)!right)!timesleft(-1+9right)$$
                  $$5985 = left(left(2+0!right)!+1right)!+9!!$$
                  $$6471 = left(left(left(2+0!right)!right)!-1right)times9$$
                  $$6480 = left(left(2+0!right)!right)!times1times9$$
                  $$6489 = left(left(left(2+0!right)!right)!+1right)times9$$
                  $$6561 = left(2+0!right)^{-1+9}$$
                  $$6615 = left(left(2+0!right)!+1right)times9!!$$
                  $$6720 = frac{left(left(2+0!+1right)!!right)!}{left(sqrt{9}right)!}$$
                  $$6859 = left(20-1right)^{sqrt{9}}$$
                  $$7200 = left(left(2+0!right)!right)!timesleft(1+9right)$$
                  $$7560 = left(2+0!+1right)!!times9!!$$
                  $$7678 = 2timesleft(-0!+left(1+9right)!!right)$$
                  $$7680 = 20timesleft(-1+9right)!!$$
                  $$7682 = 2timesleft(0!+left(1+9right)!!right)$$
                  $$7776 = left(2+0!right)!^{-1+left(sqrt{9}right)!}$$
                  $$8000 = 20^{1timessqrt{9}}$$
                  $$8192 = 2timessqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                  $$9261 = left(20+1right)^{sqrt{9}}$$
                  $$9648 = 201timesleft(left(sqrt{9}right)!right)!!$$






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                  I wrote a program to determine all representable numbers between 1 and 10000 following the rules, so this should be a comprehensive list.



                  0 through 30:




                  $$0 = 2times0timesleft(1+9right)$$
                  $$1 = 20-19$$
                  $$2 = 2+0timesleft(1+9right)$$
                  $$3 = 2+0+1^{9}$$
                  $$4 = 2-left(0+1-sqrt{9}right)$$
                  $$5 = frac{20}{1+sqrt{9}}$$
                  $$6 = -2-left(0+1-9right)$$
                  $$7 = 2timesleft(0-1right)+9$$
                  $$8 = 2times0-left(1-9right)$$
                  $$9 = 2times0+1times9$$
                  $$10 = 2-left(0+1-9right)$$
                  $$11 = 2+0+1times9$$
                  $$12 = 2+0+1+9$$
                  $$13 = 20-left(1+left(sqrt{9}right)!right)$$
                  $$14 = 20-1timesleft(sqrt{9}right)!$$
                  $$15 = 20+1-left(sqrt{9}right)!$$
                  $$16 = 2timesleft(0-left(1-9right)right)$$
                  $$17 = -2+0+19$$
                  $$18 = 2timesleft(0+1times9right)$$
                  $$19 = 2times0+19$$
                  $$20 = 2timesleft(0+1+9right)$$
                  $$21 = 2+0+19$$
                  $$22 = 20-left(1-sqrt{9}right)$$
                  $$23 = 20+1timessqrt{9}$$
                  $$24 = 20+1+sqrt{9}$$
                  $$25 = 20-left(1-left(sqrt{9}right)!right)$$
                  $$26 = 20+1timesleft(sqrt{9}right)!$$
                  $$27 = left(2+0+1right)times9$$
                  $$28 = 20-left(1-9right)$$
                  $$29 = 20+1times9$$
                  $$30 = 20+1+9$$




                  31 through 100. Interestingly enough, 31 is the smallest number that cannot be done.




                  $$32 = sqrt{2^{0+1+9}}$$
                  $$33 = left(2+0!+1right)!+9$$
                  $$35 = 20+left(-1+left(sqrt{9}right)!right)!!$$
                  $$36 = 2timesleft(-0!+19right)$$
                  $$38 = 2timesleft(0+19right)$$
                  $$39 = 20+19$$
                  $$40 = 20timesleft(-1+sqrt{9}right)$$
                  $$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt{9}right)!right)$$
                  $$42 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)$$
                  $$43 = left(left(2+0!right)!right)!!+1-left(sqrt{9}right)!$$
                  $$44 = 20+left(1+sqrt{9}right)!$$
                  $$45 = left(left(2+0!right)!-1right)times9$$
                  $$46 = 2timesleft(-0!+left(1+sqrt{9}right)!right)$$
                  $$47 = 2times0-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$48 = 2timesleft(0+left(1+sqrt{9}right)!right)$$
                  $$49 = 2-left(0+1-left(left(sqrt{9}right)!right)!!right)$$
                  $$50 = 2timesleft(0!+left(1+sqrt{9}right)!right)$$
                  $$51 = 2+0+1+left(left(sqrt{9}right)!right)!!$$
                  $$52 = 2+0!+1+left(left(sqrt{9}right)!right)!!$$
                  $$53 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$54 = left(2+0!right)!times1times9$$
                  $$55 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!!$$
                  $$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
                  $$57 = left(20-1right)timessqrt{9}$$
                  $$58 = left(left(2+0!right)!right)!!+1+9$$
                  $$60 = 20times1timessqrt{9}$$
                  $$62 = -2+left(0!+1right)^{left(sqrt{9}right)!}$$
                  $$63 = left(20+1right)timessqrt{9}$$
                  $$64 = 2^{0+1timesleft(sqrt{9}right)!}$$
                  $$66 = 2+left(0!+1right)^{left(sqrt{9}right)!}$$
                  $$67 = frac{201}{sqrt{9}}$$
                  $$68 = 20+1timesleft(left(sqrt{9}right)!right)!!$$
                  $$69 = -2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                  $$71 = sqrt{2^{0}+left(1+left(sqrt{9}right)!right)!}$$
                  $$72 = left(2+0!right)timesleft(1+sqrt{9}right)!$$
                  $$73 = 2+sqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                  $$80 = 20timesleft(1+sqrt{9}right)$$
                  $$81 = left(2+0!right)^{1+sqrt{9}}$$
                  $$85 = -20+left(1+left(sqrt{9}right)!right)!!$$
                  $$90 = frac{left(left(2+0!right)!right)!}{-1+9}$$
                  $$92 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
                  $$94 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!!right)right)$$
                  $$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$96 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!!right)$$
                  $$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt{9}right)!right)!!$$
                  $$98 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!!right)$$
                  $$99 = left(left(2+0!right)!+1right)!!-left(sqrt{9}right)!$$
                  $$100 = 20timesleft(-1+left(sqrt{9}right)!right)$$




                  101 through 1000:




                  $$102 = left(left(2+0!right)!+1right)!!-sqrt{9}$$
                  $$103 = -2+0+left(1+left(sqrt{9}right)!right)!!$$
                  $$104 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!!$$
                  $$105 = 2times0+left(1+left(sqrt{9}right)!right)!!$$
                  $$106 = 2^{0}+left(1+left(sqrt{9}right)!right)!!$$
                  $$107 = 2+0+left(1+left(sqrt{9}right)!right)!!$$
                  $$108 = 2+0!+left(1+left(sqrt{9}right)!right)!!$$
                  $$111 = left(left(2+0!right)!-1right)!-9$$
                  $$114 = left(2+0!right)!times19$$
                  $$117 = left(left(2+0!right)!-1right)!-sqrt{9}$$
                  $$118 = -2+0+left(-1+left(sqrt{9}right)!right)!$$
                  $$119 = -left(2^{0}right)+left(-1+left(sqrt{9}right)!right)!$$
                  $$120 = 20times1timesleft(sqrt{9}right)!$$
                  $$121 = 2^{0}+left(-1+left(sqrt{9}right)!right)!$$
                  $$122 = 2+0+left(-1+left(sqrt{9}right)!right)!$$
                  $$123 = left(left(2+0!right)!-1right)!+sqrt{9}$$
                  $$125 = left(left(2+0!right)!-1right)^{sqrt{9}}$$
                  $$126 = left(20+1right)timesleft(sqrt{9}right)!$$
                  $$128 = 2^{0+1+left(sqrt{9}right)!}$$
                  $$129 = left(left(2+0!right)!-1right)!+9$$
                  $$135 = left(left(2+0!right)!-1right)!!times9$$
                  $$140 = 20timesleft(1+left(sqrt{9}right)!right)$$
                  $$141 = left(left(left(2+0!right)!right)!!-1right)timessqrt{9}$$
                  $$142 = 2timessqrt{0!+left(1+left(sqrt{9}right)!right)!}$$
                  $$144 = left(2+0!right)!timesleft(1+sqrt{9}right)!$$
                  $$147 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$153 = 201-left(left(sqrt{9}right)!right)!!$$
                  $$160 = 20timesleft(-1+9right)$$
                  $$168 = left(left(2+0!right)!-1right)!+left(left(sqrt{9}right)!right)!!$$
                  $$171 = left(20-1right)times9$$
                  $$180 = 20times1times9$$
                  $$189 = left(20+1right)times9$$
                  $$192 = 201-9$$
                  $$195 = 201-left(sqrt{9}right)!$$
                  $$198 = 201-sqrt{9}$$
                  $$200 = 20timesleft(1+9right)$$
                  $$204 = 201+sqrt{9}$$
                  $$207 = 201+left(sqrt{9}right)!$$
                  $$208 = 2timesleft(-0!+left(1+left(sqrt{9}right)!right)!!right)$$
                  $$210 = 201+9$$
                  $$212 = 2timesleft(0!+left(1+left(sqrt{9}right)!right)!!right)$$
                  $$216 = left(2+0!+1right)!times9$$
                  $$224 = -left(left(2+0!right)!right)!-left(1-9!!right)$$
                  $$225 = -left(left(2+0!right)!right)!+1times9!!$$
                  $$226 = -left(left(2+0!right)!right)!+1+9!!$$
                  $$238 = 2timesleft(-0!+left(-1+left(sqrt{9}right)!right)!right)$$
                  $$240 = 2timesleft(0+left(-1+left(sqrt{9}right)!right)!right)$$
                  $$242 = 2timesleft(0!+left(-1+left(sqrt{9}right)!right)!right)$$
                  $$243 = left(2+0!right)^{-1+left(sqrt{9}right)!}$$
                  $$249 = 201+left(left(sqrt{9}right)!right)!!$$
                  $$256 = 2^{0-left(1-9right)}$$
                  $$282 = left(left(left(2+0!right)!right)!!-1right)timesleft(sqrt{9}right)!$$
                  $$288 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!!$$
                  $$294 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$300 = 20timesleft(-1+left(sqrt{9}right)!right)!!$$
                  $$315 = left(2+0!right)^{-1}times9!!$$
                  $$336 = left(left(2+0!right)!right)!-left(-1+9right)!!$$
                  $$343 = left(left(2+0!right)!+1right)^{sqrt{9}}$$
                  $$360 = 2^{0-1}timesleft(left(sqrt{9}right)!right)!$$
                  $$364 = -20+left(-1+9right)!!$$
                  $$375 = left(left(2+0!+1right)!!right)!!-9$$
                  $$378 = -left(2+0!right)!+left(-1+9right)!!$$
                  $$380 = 20times19$$
                  $$381 = -2-left(0!-left(-1+9right)!!right)$$
                  $$382 = -2+0+left(-1+9right)!!$$
                  $$383 = -left(2^{0}right)+left(-1+9right)!!$$
                  $$384 = 2times0+left(-1+9right)!!$$
                  $$385 = 2^{0}+left(-1+9right)!!$$
                  $$386 = 2+0+left(-1+9right)!!$$
                  $$387 = 2+0!+left(-1+9right)!!$$
                  $$390 = left(2+0!right)!+left(-1+9right)!!$$
                  $$393 = left(left(2+0!+1right)!!right)!!+9$$
                  $$400 = 20^{-1+sqrt{9}}$$
                  $$404 = 20+left(-1+9right)!!$$
                  $$423 = left(left(left(2+0!right)!right)!!-1right)times9$$
                  $$432 = left(left(2+0!right)!right)!!times1times9$$
                  $$441 = left(left(left(2+0!right)!right)!!+1right)times9$$
                  $$472 = 2^{-0!}timesleft(-1+9!!right)$$
                  $$473 = 2^{-0!}timesleft(1+9!!right)$$
                  $$480 = 20timesleft(1+sqrt{9}right)!$$
                  $$504 = left(left(2+0!right)!right)!^{-1}times9!$$
                  $$510 = -2+left(0!+1right)^{9}$$
                  $$512 = 2^{0+1times9}$$
                  $$514 = 2+left(0!+1right)^{9}$$
                  $$519 = -201+left(left(sqrt{9}right)!right)!$$
                  $$560 = frac{left(left(2+0!right)!+1right)!}{9}$$
                  $$561 = -left(left(2+0!+1right)!!right)!!+9!!$$
                  $$600 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!$$
                  $$603 = 201timessqrt{9}$$
                  $$615 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)!!$$
                  $$630 = left(2+0!right)!timesleft(1+left(sqrt{9}right)!right)!!$$
                  $$671 = left(left(2+0!right)!right)!-left(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$672 = left(left(2+0!right)!right)!-1timesleft(left(sqrt{9}right)!right)!!$$
                  $$673 = left(left(2+0!right)!right)!+1-left(left(sqrt{9}right)!right)!!$$
                  $$696 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)!$$
                  $$699 = -20-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$700 = -20+1timesleft(left(sqrt{9}right)!right)!$$
                  $$701 = left(left(2+0!right)!right)!-19$$
                  $$705 = left(left(2+0!right)!right)!-left(-1+left(sqrt{9}right)!right)!!$$
                  $$710 = left(left(2+0!right)!right)!-left(1+9right)$$
                  $$711 = left(left(2+0!right)!right)!-1times9$$
                  $$712 = left(left(2+0!right)!right)!+1-9$$
                  $$713 = left(left(2+0!right)!right)!-left(1+left(sqrt{9}right)!right)$$
                  $$714 = left(left(2+0!right)!right)!-1timesleft(sqrt{9}right)!$$
                  $$715 = left(left(2+0!right)!right)!+1-left(sqrt{9}right)!$$
                  $$716 = left(left(2+0!right)!right)!-left(1+sqrt{9}right)$$
                  $$717 = -2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
                  $$718 = 2timesleft(0-1right)+left(left(sqrt{9}right)!right)!$$
                  $$719 = 2times0-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$720 = 2times0+1timesleft(left(sqrt{9}right)!right)!$$
                  $$721 = 2-left(0+1-left(left(sqrt{9}right)!right)!right)$$
                  $$722 = 2+0+1timesleft(left(sqrt{9}right)!right)!$$
                  $$723 = 2+0+1+left(left(sqrt{9}right)!right)!$$
                  $$724 = 2+0!+1+left(left(sqrt{9}right)!right)!$$
                  $$725 = left(2+0!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$726 = left(2+0!right)!+1timesleft(left(sqrt{9}right)!right)!$$
                  $$727 = left(2+0!right)!+1+left(left(sqrt{9}right)!right)!$$
                  $$728 = left(left(2+0!right)!right)!-left(1-9right)$$
                  $$729 = left(2+0+1right)^{left(sqrt{9}right)!}$$
                  $$730 = left(left(2+0!right)!right)!+1+9$$
                  $$735 = left(left(2+0!right)!-1right)!!+left(left(sqrt{9}right)!right)!$$
                  $$739 = 20-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$740 = 20+1timesleft(left(sqrt{9}right)!right)!$$
                  $$741 = 20+1+left(left(sqrt{9}right)!right)!$$
                  $$744 = left(left(2+0!right)!right)!+left(1+sqrt{9}right)!$$
                  $$766 = 2timesleft(-0!+left(-1+9right)!!right)$$
                  $$767 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!!right)$$
                  $$768 = 2timesleft(0+left(-1+9right)!!right)$$
                  $$769 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!!$$
                  $$770 = 2timesleft(0!+left(-1+9right)!!right)$$
                  $$825 = -left(left(2+0!right)!-1right)!+9!!$$
                  $$840 = frac{left(left(2+0!right)!+1right)!}{left(sqrt{9}right)!}$$
                  $$896 = -left(left(2+0!right)!right)!!-left(1-9!!right)$$
                  $$897 = -left(left(2+0!right)!right)!!+1times9!!$$
                  $$898 = -left(left(2+0!right)!right)!!+1+9!!$$
                  $$912 = left(left(2+0!right)!right)!!times19$$
                  $$921 = 201+left(left(sqrt{9}right)!right)!$$
                  $$924 = -20-left(1-9!!right)$$
                  $$925 = -20+1times9!!$$
                  $$926 = -20+1+9!!$$
                  $$930 = -left(left(2+0!right)!-1right)!!+9!!$$
                  $$937 = -left(2+0!+1right)!!+9!!$$
                  $$938 = -left(2+0!right)!-left(1-9!!right)$$
                  $$939 = -left(2+0!right)!+1times9!!$$
                  $$940 = 20timesleft(-1+left(left(sqrt{9}right)!right)!!right)$$
                  $$941 = -2-left(0!+1-9!!right)$$
                  $$942 = -2-left(0+1-9!!right)$$
                  $$943 = 2timesleft(0-1right)+9!!$$
                  $$944 = 2times0-left(1-9!!right)$$
                  $$945 = 2times0+1times9!!$$
                  $$946 = 2-left(0+1-9!!right)$$
                  $$947 = 2+0+1times9!!$$
                  $$948 = 2+0+1+9!!$$
                  $$949 = 2+0!+1+9!!$$
                  $$950 = left(2+0!right)!-left(1-9!!right)$$
                  $$951 = left(2+0!right)!+1times9!!$$
                  $$952 = left(2+0!right)!+1+9!!$$
                  $$953 = left(2+0!+1right)!!+9!!$$
                  $$960 = 20times1timesleft(left(sqrt{9}right)!right)!!$$
                  $$964 = 20-left(1-9!!right)$$
                  $$965 = 20+1times9!!$$
                  $$966 = 20+1+9!!$$
                  $$969 = left(2+0!+1right)!+9!!$$
                  $$980 = 20timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$992 = left(left(2+0!right)!right)!!-left(1-9!!right)$$
                  $$993 = left(left(2+0!right)!right)!!+1times9!!$$
                  $$994 = left(left(2+0!right)!right)!!+1+9!!$$




                  1001 through 10000:




                  $$1008 = left(20+1right)timesleft(left(sqrt{9}right)!right)!!$$
                  $$1024 = 2^{0+1+9}$$
                  $$1050 = left(left(2+0!right)!+1right)!!+9!!$$
                  $$1065 = left(left(2+0!right)!-1right)!+9!!$$
                  $$1080 = left(left(2+0!right)!-1right)!times9$$
                  $$1104 = left(left(2+0!right)!right)!+left(-1+9right)!!$$
                  $$1146 = 201+9!!$$
                  $$1152 = left(2+0!right)timesleft(-1+9right)!!$$
                  $$1206 = 201timesleft(sqrt{9}right)!$$
                  $$1296 = left(2+0!right)!^{1+sqrt{9}}$$
                  $$1329 = left(left(2+0!+1right)!!right)!!+9!!$$
                  $$1436 = 2timesleft(-0!-left(1-left(left(sqrt{9}right)!right)!right)right)$$
                  $$1438 = 2timesleft(0-left(1-left(left(sqrt{9}right)!right)!right)right)$$
                  $$1439 = left(left(2+0!right)!right)!-left(1-left(left(sqrt{9}right)!right)!right)$$
                  $$1440 = 2timesleft(0+1timesleft(left(sqrt{9}right)!right)!right)$$
                  $$1441 = left(left(2+0!right)!right)!+1+left(left(sqrt{9}right)!right)!$$
                  $$1442 = 2timesleft(0+1+left(left(sqrt{9}right)!right)!right)$$
                  $$1444 = 2timesleft(0!+1+left(left(sqrt{9}right)!right)!right)$$
                  $$1664 = left(left(2+0!right)!right)!-left(1-9!!right)$$
                  $$1665 = left(left(2+0!right)!right)!+1times9!!$$
                  $$1666 = left(left(2+0!right)!right)!+1+9!!$$
                  $$1680 = frac{left(left(2+0!right)!+1right)!}{sqrt{9}}$$
                  $$1809 = 201times9$$
                  $$1886 = 2timesleft(-0!-left(1-9!!right)right)$$
                  $$1888 = 2timesleft(0-left(1-9!!right)right)$$
                  $$1890 = 2timesleft(0+1times9!!right)$$
                  $$1892 = 2timesleft(0+1+9!!right)$$
                  $$1894 = 2timesleft(0!+1+9!!right)$$
                  $$1920 = 2^{-0!}timesleft(1+9right)!!$$
                  $$2019 = 2019$$
                  $$2048 = 2^{0!+1+9}$$
                  $$2100 = 20timesleft(1+left(sqrt{9}right)!right)!!$$
                  $$2145 = frac{left(left(left(2+0!right)!-1right)!!right)!!}{9!!}$$
                  $$2157 = left(left(left(2+0!right)!right)!-1right)timessqrt{9}$$
                  $$2160 = left(2+0+1right)timesleft(left(sqrt{9}right)!right)!$$
                  $$2163 = left(2+0!right)timesleft(1+left(left(sqrt{9}right)!right)!right)$$
                  $$2187 = left(2+0!right)^{1+left(sqrt{9}right)!}$$
                  $$2256 = left(left(left(2+0!right)!right)!!-1right)timesleft(left(sqrt{9}right)!right)!!$$
                  $$2304 = left(2+0!right)!timesleft(-1+9right)!!$$
                  $$2352 = left(left(2+0!right)!right)!!timesleft(1+left(left(sqrt{9}right)!right)!!right)$$
                  $$2400 = 20timesleft(-1+left(sqrt{9}right)!right)!$$
                  $$2520 = 2^{-0!}timesleft(1+left(sqrt{9}right)!right)!$$
                  $$2832 = left(2+0!right)timesleft(-1+9!!right)$$
                  $$2835 = left(2+0+1right)times9!!$$
                  $$2838 = left(2+0!right)timesleft(1+9!!right)$$
                  $$2880 = 2timesleft(0!+1right)timesleft(left(sqrt{9}right)!right)!$$
                  $$3120 = -left(left(2+0!right)!right)!+left(1+9right)!!$$
                  $$3375 = left(left(2+0!right)!-1right)!!^{sqrt{9}}$$
                  $$3456 = left(left(2+0!+1right)!!right)!!times9$$
                  $$3600 = left(left(2+0!right)!-1right)timesleft(left(sqrt{9}right)!right)!$$
                  $$3780 = 2timesleft(0!+1right)times9!!$$
                  $$3792 = -left(left(2+0!right)!right)!!+left(1+9right)!!$$
                  $$3820 = -20+left(1+9right)!!$$
                  $$3834 = -left(2+0!right)!+left(1+9right)!!$$
                  $$3837 = -2-left(0!-left(1+9right)!!right)$$
                  $$3838 = -2+0+left(1+9right)!!$$
                  $$3839 = -left(2^{0}right)+left(1+9right)!!$$
                  $$3840 = 2times0+left(1+9right)!!$$
                  $$3841 = 2^{0}+left(1+9right)!!$$
                  $$3842 = 2+0+left(1+9right)!!$$
                  $$3843 = 2+0!+left(1+9right)!!$$
                  $$3846 = left(2+0!right)!+left(1+9right)!!$$
                  $$3860 = 20+left(1+9right)!!$$
                  $$3888 = left(left(2+0!right)!right)!!+left(1+9right)!!$$
                  $$4094 = -2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                  $$4095 = left(left(2+0!right)!+1right)!-9!!$$
                  $$4096 = 2^{left(0!+1right)timesleft(sqrt{9}right)!}$$
                  $$4098 = 2+sqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                  $$4314 = left(left(left(2+0!right)!right)!-1right)timesleft(sqrt{9}right)!$$
                  $$4320 = left(2+0!right)!times1timesleft(left(sqrt{9}right)!right)!$$
                  $$4326 = left(2+0!right)!timesleft(1+left(left(sqrt{9}right)!right)!right)$$
                  $$4480 = frac{left(left(2+0!+1right)!!right)!}{9}$$
                  $$4560 = left(left(2+0!right)!right)!+left(1+9right)!!$$
                  $$4725 = left(left(2+0!right)!-1right)times9!!$$
                  $$4992 = left(left(2+0!right)!+1right)!-left(left(sqrt{9}right)!right)!!$$
                  $$5020 = -20+left(1+left(sqrt{9}right)!right)!$$
                  $$5031 = left(left(2+0!right)!+1right)!-9$$
                  $$5034 = left(left(2+0!right)!+1right)!-left(sqrt{9}right)!$$
                  $$5037 = left(left(2+0!right)!+1right)!-sqrt{9}$$
                  $$5038 = -2+0+left(1+left(sqrt{9}right)!right)!$$
                  $$5039 = -left(2^{0}right)+left(1+left(sqrt{9}right)!right)!$$
                  $$5040 = 2times0+left(1+left(sqrt{9}right)!right)!$$
                  $$5041 = 2^{0}+left(1+left(sqrt{9}right)!right)!$$
                  $$5042 = 2+0+left(1+left(sqrt{9}right)!right)!$$
                  $$5043 = 2+0!+left(1+left(sqrt{9}right)!right)!$$
                  $$5046 = left(2+0!right)!+left(1+left(sqrt{9}right)!right)!$$
                  $$5049 = left(left(2+0!right)!+1right)!+9$$
                  $$5060 = 20+left(1+left(sqrt{9}right)!right)!$$
                  $$5088 = left(left(2+0!right)!right)!!+left(1+left(sqrt{9}right)!right)!$$
                  $$5664 = left(2+0!right)!timesleft(-1+9!!right)$$
                  $$5670 = left(2+0!right)!times1times9!!$$
                  $$5676 = left(2+0!right)!timesleft(1+9!!right)$$
                  $$5760 = left(left(2+0!right)!right)!timesleft(-1+9right)$$
                  $$5985 = left(left(2+0!right)!+1right)!+9!!$$
                  $$6471 = left(left(left(2+0!right)!right)!-1right)times9$$
                  $$6480 = left(left(2+0!right)!right)!times1times9$$
                  $$6489 = left(left(left(2+0!right)!right)!+1right)times9$$
                  $$6561 = left(2+0!right)^{-1+9}$$
                  $$6615 = left(left(2+0!right)!+1right)times9!!$$
                  $$6720 = frac{left(left(2+0!+1right)!!right)!}{left(sqrt{9}right)!}$$
                  $$6859 = left(20-1right)^{sqrt{9}}$$
                  $$7200 = left(left(2+0!right)!right)!timesleft(1+9right)$$
                  $$7560 = left(2+0!+1right)!!times9!!$$
                  $$7678 = 2timesleft(-0!+left(1+9right)!!right)$$
                  $$7680 = 20timesleft(-1+9right)!!$$
                  $$7682 = 2timesleft(0!+left(1+9right)!!right)$$
                  $$7776 = left(2+0!right)!^{-1+left(sqrt{9}right)!}$$
                  $$8000 = 20^{1timessqrt{9}}$$
                  $$8192 = 2timessqrt{sqrt{left(0!+1right)^{left(left(sqrt{9}right)!right)!!}}}$$
                  $$9261 = left(20+1right)^{sqrt{9}}$$
                  $$9648 = 201timesleft(left(sqrt{9}right)!right)!!$$







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                      Quarter-circle Tiles

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                      up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
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                      build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

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                      0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
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                      Mont Emei

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                      Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
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