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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

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  • $\begingroup$ is this something you composed yourself? $\endgroup$
    – Kate Gregory
    Nov 26, 2018 at 15:13
  • $\begingroup$ @KateGregory: A variation on an old theme. $\endgroup$
    – Bernardo Recamán Santos
    Nov 26, 2018 at 15:15
  • $\begingroup$ Zero (as a square) allowed? $\endgroup$
    – z100
    Nov 26, 2018 at 20:43
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    $\begingroup$ @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. $\endgroup$
    – user46002
    Nov 26, 2018 at 21:55

2 Answers 2

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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

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  • $\begingroup$ 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) ; $\endgroup$
    – z100
    Nov 26, 2018 at 21:25
  • $\begingroup$ @z100 I'm afraid I'm not sure what you're asking - could you clarify? $\endgroup$
    – B. Mehta
    Nov 26, 2018 at 21:33
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    $\begingroup$ 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. $\endgroup$
    – benj2240
    Nov 27, 2018 at 1:03
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    $\begingroup$ @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. $\endgroup$
    – benj2240
    Nov 28, 2018 at 1:14
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    $\begingroup$ @benj2240: Yes, 59 is the lowest my students have achieved. $\endgroup$
    – Bernardo Recamán Santos
    Nov 28, 2018 at 1:18
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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.

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