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










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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










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










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






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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:




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







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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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      4 Answers
      4






      active

      oldest

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      4 Answers
      4






      active

      oldest

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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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                      Mont Emei