Primes and Squares

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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
    Nov 26 at 15:13










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    Nov 26 at 15:15










  • Zero (as a square) allowed?
    – z100
    Nov 26 at 20:43






  • 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
    Nov 26 at 21:55















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
    Nov 26 at 15:13










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    Nov 26 at 15:15










  • Zero (as a square) allowed?
    – z100
    Nov 26 at 20:43






  • 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
    Nov 26 at 21:55













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 Nov 26 at 14:59









Bernardo Recamán Santos

2,2881141




2,2881141











  • is this something you composed yourself?
    – Kate Gregory
    Nov 26 at 15:13










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    Nov 26 at 15:15










  • Zero (as a square) allowed?
    – z100
    Nov 26 at 20:43






  • 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
    Nov 26 at 21:55

















  • is this something you composed yourself?
    – Kate Gregory
    Nov 26 at 15:13










  • @KateGregory: A variation on an old theme.
    – Bernardo Recamán Santos
    Nov 26 at 15:15










  • Zero (as a square) allowed?
    – z100
    Nov 26 at 20:43






  • 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
    Nov 26 at 21:55
















is this something you composed yourself?
– Kate Gregory
Nov 26 at 15:13




is this something you composed yourself?
– Kate Gregory
Nov 26 at 15:13












@KateGregory: A variation on an old theme.
– Bernardo Recamán Santos
Nov 26 at 15:15




@KateGregory: A variation on an old theme.
– Bernardo Recamán Santos
Nov 26 at 15:15












Zero (as a square) allowed?
– z100
Nov 26 at 20:43




Zero (as a square) allowed?
– z100
Nov 26 at 20:43




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
Nov 26 at 21:55





@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
Nov 26 at 21:55











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






















  • 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
    Nov 26 at 21:25











  • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
    – B. Mehta
    Nov 26 at 21:33






  • 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
    Nov 27 at 1:03






  • 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
    Nov 28 at 1:14






  • 1




    @benj2240: Yes, 59 is the lowest my students have achieved.
    – Bernardo Recamán Santos
    Nov 28 at 1:18

















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






















    • 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
      Nov 26 at 21:25











    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      Nov 26 at 21:33






    • 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
      Nov 27 at 1:03






    • 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
      Nov 28 at 1:14






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      Nov 28 at 1:18














    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






















    • 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
      Nov 26 at 21:25











    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      Nov 26 at 21:33






    • 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
      Nov 27 at 1:03






    • 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
      Nov 28 at 1:14






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      Nov 28 at 1:18












    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














    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














    share|improve this answer



    share|improve this answer








    edited Nov 26 at 21:20

























    answered Nov 26 at 20:28









    B. Mehta

    1963




    1963











    • 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
      Nov 26 at 21:25











    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      Nov 26 at 21:33






    • 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
      Nov 27 at 1:03






    • 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
      Nov 28 at 1:14






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      Nov 28 at 1:18
















    • 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
      Nov 26 at 21:25











    • @z100 I'm afraid I'm not sure what you're asking - could you clarify?
      – B. Mehta
      Nov 26 at 21:33






    • 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
      Nov 27 at 1:03






    • 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
      Nov 28 at 1:14






    • 1




      @benj2240: Yes, 59 is the lowest my students have achieved.
      – Bernardo Recamán Santos
      Nov 28 at 1:18















    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
    Nov 26 at 21:25





    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
    Nov 26 at 21:25













    @z100 I'm afraid I'm not sure what you're asking - could you clarify?
    – B. Mehta
    Nov 26 at 21:33




    @z100 I'm afraid I'm not sure what you're asking - could you clarify?
    – B. Mehta
    Nov 26 at 21:33




    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
    Nov 27 at 1:03




    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
    Nov 27 at 1:03




    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
    Nov 28 at 1:14




    @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
    Nov 28 at 1:14




    1




    1




    @benj2240: Yes, 59 is the lowest my students have achieved.
    – Bernardo Recamán Santos
    Nov 28 at 1:18




    @benj2240: Yes, 59 is the lowest my students have achieved.
    – Bernardo Recamán Santos
    Nov 28 at 1:18










    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 Nov 26 at 16:51









        gabbo1092

        4,683738




        4,683738










        answered Nov 26 at 16:32









        Excited Raichu

        4,687754




        4,687754



























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