# Primes and Squares

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.

• is this something you composed yourself? – Kate Gregory Nov 26 '18 at 15:13
• @KateGregory: A variation on an old theme. – Bernardo Recamán Santos Nov 26 '18 at 15:15
• Zero (as a square) allowed? – z100 Nov 26 '18 at 20:43
• @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 '18 at 21:55

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 

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

Alright, I’ve definitely got an upper bound 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.