I have in front of me five standard dice, touching in a row, left to right.

Treating each face as a base-10 digit with the value of its number of pips, the five dice, on their faces that are toward me, form a 5-digit prime. The tops of the dice, also taken in order, form a 5-digit perfect square.

If I were looking at these five dice from the opposite direction, I'd see a different prime number formed by the digits on the vertical faces in order, and I'd see a different perfect square formed by the tops, again taken in order, as seen. In fact, that perfect square would be larger than the one I'm seeing from the side I'm actually on.

And one more thing: the five digits of each prime are different, but of course any given digit might or might not be on both primes.

1. Identify the primes and squares involved.

2. If duplicate digits (any multiples of the same digit within a number) were allowed, what could the front and back primes be, and the resulting squares on top, other than the ones found in part 1, using the same other rules as part 1?

(Part 1)
The number on the opposing face of a standard die is equal to 77777 minus the 'reversgram' of the number on the opposing face.

Only three faces of each die are referenced. On the 'front' face is the prime 43651; on the top is the square 12544 [112²] with its reversgramic square 44521 [211²] as viewed in the opposite direction; and on the 'back' face is the reversgramic prime 62143.

(Part 2)
There are only four 5-digit squares composed of digit values 1 to 6 with reversgrams that are also squares. They are 12321, 12544, 14641 and 44521. With the restriction given that one of the squares is larger than the other, only 12544, with its reversgram of 44521, fits the requirement for the solution. (12321 and 14641 are both palindromes, thus their reversgrams are neither greater nor less than but equal to themselves.)

With digits on the primes unrestricted in duplication, with the top of the dice being 12544 (and its reversgram 44521), excluding the solution to part 1, the possible pairs of corresponding facings that are the primes are 41611 & 66163 and 43411 & 66343.

Posted by Dej Mar on 2011-09-10 13:48:40