Primes and Squares











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7
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Place a different prime or square number on each of the fifteen disks below so that the number in any disk that lies on two others is the sum of the numbers in those disks. Do so in such a way that the number on the apex is as small as possible.



enter image description here






mathematics arithmetic







share|improve this question






















  • is this something you composed yourself?
    – Kate Gregory
    2 days ago










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    2 days ago










  • Zero (as a square) allowed?
    – z100
    2 days ago






  • 1




    @z100 You could not use a zero, since $x + 0 = x$, and therefore you'd have to have two $x$'es in your grid; that is disallowed. See imgur.com/a/gPWWkaN for explanation.
    – Hugh
    2 days ago

















up vote
7
down vote

favorite












Place a different prime or square number on each of the fifteen disks below so that the number in any disk that lies on two others is the sum of the numbers in those disks. Do so in such a way that the number on the apex is as small as possible.



enter image description here






mathematics arithmetic







share|improve this question






















  • is this something you composed yourself?
    – Kate Gregory
    2 days ago










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    2 days ago










  • Zero (as a square) allowed?
    – z100
    2 days ago






  • 1




    @z100 You could not use a zero, since $x + 0 = x$, and therefore you'd have to have two $x$'es in your grid; that is disallowed. See imgur.com/a/gPWWkaN for explanation.
    – Hugh
    2 days ago















up vote
7
down vote

favorite









up vote
7
down vote

favorite











Place a different prime or square number on each of the fifteen disks below so that the number in any disk that lies on two others is the sum of the numbers in those disks. Do so in such a way that the number on the apex is as small as possible.



enter image description here






mathematics arithmetic







share|improve this question













Place a different prime or square number on each of the fifteen disks below so that the number in any disk that lies on two others is the sum of the numbers in those disks. Do so in such a way that the number on the apex is as small as possible.



enter image description here






mathematics arithmetic




mathematics arithmetic






share|improve this question













share|improve this question











share|improve this question




share|improve this question










asked 2 days ago









Bernardo Recamán Santos

2,1781139




2,1781139












  • is this something you composed yourself?
    – Kate Gregory
    2 days ago










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    2 days ago










  • Zero (as a square) allowed?
    – z100
    2 days ago






  • 1




    @z100 You could not use a zero, since $x + 0 = x$, and therefore you'd have to have two $x$'es in your grid; that is disallowed. See imgur.com/a/gPWWkaN for explanation.
    – Hugh
    2 days ago




















  • is this something you composed yourself?
    – Kate Gregory
    2 days ago










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    2 days ago










  • Zero (as a square) allowed?
    – z100
    2 days ago






  • 1




    @z100 You could not use a zero, since $x + 0 = x$, and therefore you'd have to have two $x$'es in your grid; that is disallowed. See imgur.com/a/gPWWkaN for explanation.
    – Hugh
    2 days ago


















is this something you composed yourself?
– Kate Gregory
2 days ago




is this something you composed yourself?
– Kate Gregory
2 days ago












@KateGregory: A variation on an old theme.
– Bernardo Recamán Santos
2 days ago




@KateGregory: A variation on an old theme.
– Bernardo Recamán Santos
2 days ago












Zero (as a square) allowed?
– z100
2 days ago




Zero (as a square) allowed?
– z100
2 days ago




1




1




@z100 You could not use a zero, since $x + 0 = x$, and therefore you'd have to have two $x$'es in your grid; that is disallowed. See imgur.com/a/gPWWkaN for explanation.
– Hugh
2 days ago






@z100 You could not use a zero, since $x + 0 = x$, and therefore you'd have to have two $x$'es in your grid; that is disallowed. See imgur.com/a/gPWWkaN for explanation.
– Hugh
2 days ago












2 Answers
2






active

oldest

votes

















up vote
8
down vote



accepted










A much lower upper bound, which I'm fairly sure is optimal (assuming 0 is disallowed).





              1669 
           576 || 1093 
       383 || 193 || 900 
    347 ||  36 || 157 || 743 
 324 || 23 ||  13 || 144 || 599






share|improve this answer










New contributor




B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.


















  • Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
    – z100
    2 days ago












  • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
    – B. Mehta
    2 days ago






  • 1




    My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
    – benj2240
    2 days ago






  • 1




    @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
    – benj2240
    yesterday






  • 1




    @benj2240: Yes, 59 is the lowest my students have achieved.
    – Bernardo Recamán Santos
    yesterday


















up vote
5
down vote













Alright, I’ve definitely got an upper bound here.




enter image description here




In text:





                  390625 
             140625 || 250000 
         50625 || 90000 || 160000 
     18225 || 32400 || 57600 || 102400 
 6561 || 11664 || 20736 || 36864 || 65536



However,




this uses all square numbers, and is far from optimal. I’ll have to see if I can reduce it by using primes.







share|improve this answer























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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    8
    down vote



    accepted










    A much lower upper bound, which I'm fairly sure is optimal (assuming 0 is disallowed).





                  1669 
               576 || 1093 
           383 || 193 || 900 
        347 ||  36 || 157 || 743 
     324 || 23 ||  13 || 144 || 599






    share|improve this answer










    New contributor




    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.


















    • Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
      – z100
      2 days ago












    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      2 days ago






    • 1




      My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
      – benj2240
      2 days ago






    • 1




      @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
      – benj2240
      yesterday






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      yesterday















    up vote
    8
    down vote



    accepted










    A much lower upper bound, which I'm fairly sure is optimal (assuming 0 is disallowed).





                  1669 
               576 || 1093 
           383 || 193 || 900 
        347 ||  36 || 157 || 743 
     324 || 23 ||  13 || 144 || 599






    share|improve this answer










    New contributor




    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.


















    • Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
      – z100
      2 days ago












    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      2 days ago






    • 1




      My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
      – benj2240
      2 days ago






    • 1




      @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
      – benj2240
      yesterday






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      yesterday













    up vote
    8
    down vote



    accepted







    up vote
    8
    down vote



    accepted






    A much lower upper bound, which I'm fairly sure is optimal (assuming 0 is disallowed).





                  1669 
               576 || 1093 
           383 || 193 || 900 
        347 ||  36 || 157 || 743 
     324 || 23 ||  13 || 144 || 599






    share|improve this answer










    New contributor




    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    A much lower upper bound, which I'm fairly sure is optimal (assuming 0 is disallowed).





                  1669 
               576 || 1093 
           383 || 193 || 900 
        347 ||  36 || 157 || 743 
     324 || 23 ||  13 || 144 || 599







    share|improve this answer










    New contributor




    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    share|improve this answer



    share|improve this answer








    edited 2 days ago





















    New contributor




    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    answered 2 days ago









    B. Mehta

    1963




    1963




    New contributor




    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.





    New contributor





    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    B. Mehta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.












    • Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
      – z100
      2 days ago












    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      2 days ago






    • 1




      My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
      – benj2240
      2 days ago






    • 1




      @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
      – benj2240
      yesterday






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      yesterday


















    • Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
      – z100
      2 days ago












    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      2 days ago






    • 1




      My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
      – benj2240
      2 days ago






    • 1




      @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
      – benj2240
      yesterday






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      yesterday
















    Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
    – z100
    2 days ago






    Algorithm used extening the order? E.G.: 1st order: 1 ; 2nd order: 3 (1 2) ; 3rd order: 16 (3 13) (1 2 11) ; or 16 (13 3) (12 1 2) ;
    – z100
    2 days ago














    @z100 I'm afraid I'm not sure what you're asking - could you clarify?
    – B. Mehta
    2 days ago




    @z100 I'm afraid I'm not sure what you're asking - could you clarify?
    – B. Mehta
    2 days ago




    1




    1




    My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
    – benj2240
    2 days ago




    My own (extremely shameful, dirty, brute-force) code confirms this answer is optimal. I can also provide the smallest apex value for a 4-level tree, which is 23.
    – benj2240
    2 days ago




    1




    1




    @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
    – benj2240
    yesterday




    @BernardoRecamánSantos Oh! You're right, 23 isn't possible. I didn't notice that I had a duplicated 3... Dirty code leads to dirty bugs. Let me correct myself: The smallest apex value for a 4-level tree is 59.
    – benj2240
    yesterday




    1




    1




    @benj2240: Yes, 59 is the lowest my students have achieved.
    – Bernardo Recamán Santos
    yesterday




    @benj2240: Yes, 59 is the lowest my students have achieved.
    – Bernardo Recamán Santos
    yesterday










    up vote
    5
    down vote













    Alright, I’ve definitely got an upper bound here.




    enter image description here




    In text:





                      390625 
                 140625 || 250000 
             50625 || 90000 || 160000 
         18225 || 32400 || 57600 || 102400 
     6561 || 11664 || 20736 || 36864 || 65536



    However,




    this uses all square numbers, and is far from optimal. I’ll have to see if I can reduce it by using primes.







    share|improve this answer



























      up vote
      5
      down vote













      Alright, I’ve definitely got an upper bound here.




      enter image description here




      In text:





                        390625 
                   140625 || 250000 
               50625 || 90000 || 160000 
           18225 || 32400 || 57600 || 102400 
       6561 || 11664 || 20736 || 36864 || 65536



      However,




      this uses all square numbers, and is far from optimal. I’ll have to see if I can reduce it by using primes.







      share|improve this answer

























        up vote
        5
        down vote










        up vote
        5
        down vote









        Alright, I’ve definitely got an upper bound here.




        enter image description here




        In text:





                          390625 
                     140625 || 250000 
                 50625 || 90000 || 160000 
             18225 || 32400 || 57600 || 102400 
         6561 || 11664 || 20736 || 36864 || 65536



        However,




        this uses all square numbers, and is far from optimal. I’ll have to see if I can reduce it by using primes.







        share|improve this answer














        Alright, I’ve definitely got an upper bound here.




        enter image description here




        In text:





                          390625 
                     140625 || 250000 
                 50625 || 90000 || 160000 
             18225 || 32400 || 57600 || 102400 
         6561 || 11664 || 20736 || 36864 || 65536



        However,




        this uses all square numbers, and is far from optimal. I’ll have to see if I can reduce it by using primes.








        share|improve this answer














        share|improve this answer



        share|improve this answer








        edited 2 days ago









        gabbo1092

        4,673736




        4,673736










        answered 2 days ago









        Excited Raichu

        4,315752




        4,315752






























             

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