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This Python program uses the

SubstitutedExpression()solver from theenigma.pylibrary to solve the embedded problem and find possible sets that satisfy the conditions Tom, Dick and Harry are looking for, and then examines which arrangement of possible solutions satisfies the remaining conditions. It runs in 86ms.Solution:The three squares in the set none of them found were: 36, 841, 9025.There are only 4 possible sets of squares:

Dick found set [1]. Tom and Harry found sets [3] and [4]. Leaving [2] as the set none of them found.

Multiple configuration output in MiniZinc convieniently produces the four sets of squares