Replace the numerals 1 through 8 with ever increasing prime numbers, always using the next lowest possible that is available
^{1} to fulfill the criteria on the left. Then do the same for the right.
Present a series for the left, and one for the right.
7 + 8 = Square
6 = Prime
4 + 5 = Square
1 + 2 + 3 = Square


7 + 8 = Cube
6 = Prime
4 + 5 = Cube
1 + 2 + 3 = Cube

If it was required that the Right set required "1" to be the next Prime following on after that used for the "8" in the Left set, what might the Right set read, if indeed it is possible?
^{1.}
Note,
"always using the next lowest possible that is available" means that if it is next on the list it cannot be dismissed unless it is
the last of a group of two or three and will not fulfill the criterion. Only then may you advance to the next.
(In reply to
re: computer solution  query by brianjn)
They all follow the rule of taking the next available prime that meets the square or cube condition. The only variable is the choice of prime for the "1" spot on the board of squares. Everything else is determined from there.
So I'm saying that the choice of 5 for the starting "1" on the square set, inevitably leads to the lowest possibe "8" position on the cube set, that is, 3253. However, the lowest average of the 16 numbers seems to occur when you start with 13 at the "1" position on the board of squares.

Posted by Charlie
on 20100106 22:36:09 