Pat has made up a list of 5-digit perfect squares (all different) which use only the digits 0 through 4, none of them starting or ending with a zero. Each of the digits 0 through 4 is used a different number of times, those numbers of times being 5 through 9.

Mike has also chosen some 5-digit perfect squares, but using only the digits from 5 through 9. Each of his digits 5 through 9 is used a different number of times in his squares, those numbers of times being 0 through 4.

What were the eligible squares for each of Pat and Mike, based on their constituent digits? Which of them did each actually use?

I used mathematica to find all possible 5 digit squares that fit the criteria for each person, then work thru all the subsets to find one that matched the digit count criteria.  This is what I got

 

Pat: 10201,10404,12321,23104,32041,33124,40401

or 101^2,102^2,111^2,152^2,179^2,182^2,201^2

Mike: 55696,97969 or 236^2,313^2

Pat's digit count is

0:8, 1:9, 2:6, 3:5, 4:7

Mike's digit count is

5:2, 6:3, 7:1, 8:0, 9:4

Posted by Daniel on 2008-08-12 17:24:22