A small number of standard dice were thrown. The total of the spots on the tops was a perfect square. Then the digits that represent each of the top numbers were placed in a row in increasing order to spell out a rather large number (a given digit may appear more than once, as in 12224).

The same thing was repeated with twice as many dice and again the total number of pips was a perfect square and a number formed by using the digits that represent the pips in ascending order was the square of the first number formed that way.

What were the two numbers formed by the digits representing the pips (the first number and its square)?

(In reply to edge of darkness / solution by ed bottemiller)

I decided to play on a hunch, I say your answer had an interesting pattern, it consists of 7 3's then a 4, its square consists of 8 1's then 7 5's and then a 6.  I have not worked out a proof yet but it seems (tested for N<=1,000,000) that if you take a number with N 3's then a 4 and square it you get a number with N+1 1's then N 5's then a 6 and thus a number with 2*(N+1) digits and thus twice the orignial number of digits.  So I decided to see if numbers of this form would work in general.  Of course for it to work we would also need there digit sums to be both be squares and thus we need in general both 3n+4 and 6n+7 to be squares.  I ran a quick search for n<=1,000,000 and found that the next number after n=7 that works is n=279 and then n=9519, and then n=323407.  And so 8 dice in your solution suddenly seems quite small compared to the next possible solution and thus must be the answer sought.  I am still working on finding a pattern to the n's that give valid solutions but my number theory is a bit rusty so it may take a while if I find one at all.

Posted by Daniel on 2009-03-28 00:22:38