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)?

I was about to give up on this when I decided to try one further pair (8 and 16 dice) and found what I think is a solution:

33333334 sums to 25 -- for 8 digits

1111111155555556 sums to 49 -- for 16 digits

where the second is the square of the first.

(I tried all pairs from 3-6 thru 7-14 with no hits.)