Problems & Puzzles: Puzzles

Puzzle 577.- Happy prime Valentine day!

In the splendid book of Nobuyuki Yoshigahara "Puzzles 101" we found the Puzzle 66 (page 41) asking for a two-dimension rectangular array of gold circles surrounded by one layer of blue circles (as shown in the image below), such that the quantity circles of each color is the same number.

Evidently this image is NOT a solution for the condition of the puzzle, because there are 2x3=6 gold circles vs 2x(2+3)+4=14 blue circles.

Mr Yoshigahara wrote in the solution page (page 100) for this puzzle that the three-dimensional version of this puzzle has 20 distinct solutions.

Today is February 12, very near of Valentine´s day. Perhaps that helped and I saw in my mind the three-dimensional version of this puzzle as a chocolate gift to my wife Luz María.

¿Could it be that some of the 20 solutions for this kind of chocolate gift is related to prime numbers?

Let's say that the gold chocolate spheres of the inner part of the gift count p, q & r spheres in each dimension.

Q. Is there a chocolate gift, satisfying the Yoshigahara's conditions, using prime numbers in some or all p, q & r?

Contributions came from Robert D Mohr, J. K. Andersen, Jan van Delden, Farid Lian, Alexandre Patarot, Hakan Summakoğlu, A. Verroken & Emmanuel Vantieghem.

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All of them found that:

We want positive solutions to 2*p*q*r = (p+2)*(q+2)*(r+2).
The left-hand side is even. The right-hand side can only be even if p, q or r
is even.
If p=2 then the equation becomes q*r = (q+2)*(r+2) with no positive solutions.
This means there can be no solutions where all sides are prime.
The same argument applies to all higher dimensions, but it might have been
hard anyway to get 4-dimensional chocolate.
There is only a boring one-dimensional prime solution with 2 gold lines and 2
blue lines at the ends.

A computer search for all 3-dimensional solutions gives:

p   q   r  p*q*r
3  11 130  4290  (two primes)
3  12  70  2520
3  13  50  1950  (two primes)
3  14  40  1680
3  15  34  1530
3  16  30  1440
3  18  25  1350
3  20  22  1320
4   7  54  1512
4   8  30   960
4   9  22   792
4  10  18   720
4  12  14   672
5   5  98  2450  (two identical primes)
5   6  28   840
5   7  18   630  (two primes)
5   8  14   560
6   6  16   576
6   7  12   504
6   8  10   480

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