(In reply to re(3): Solution (as interpreted) by brianjn)

I had arrived at the same solution(s) as ed and Charlie. But, due to the large number of possible solutions, I tried to see if there could be an interpretation that might limit the bulk to a smaller, if not unique, set.

With the assumption that a solution might be sought such that each digit appeared only once in the set of bases, I arrived at the following:

• 16₁₄, 36₃₆, 49₂₀, 64₅₇, 100₉₈
• 16₁₄, 36₃₆, 49₂₀, 64₅₈, 100₉₇
• 16₁₆, 36₃₀, 49₂₄, 64₅₇, 100₉₈
• 16₁₆, 36₃₀, 49₂₄, 64₅₈, 100₉₇
• 16₁₆, 36₃₄, 49₂₀, 64₅₇, 100₉₈
• 16₁₆, 36₃₄, 49₂₀, 64₅₈, 100₉₇

The sum of the bases add to the perfect square 225 (15²).
Of these there are two sets with the minimum total of their base 10 values. This total is 76. (The other four sets total 80 each):

• 16₁₆=10, 36₃₀=16, 49₂₄=21, 64₅₇=17, 100₉₈=12
• 16₁₆=10, 36₃₀=16, 49₂₄=21, 64₅₈=16, 100₉₇=13

Posted by Dej Mar on 2010-07-07 08:16:52