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75 different integer numbers are written on a blackboard. Each is erased and replaced with either its square or its cube, the operation being random for each. What is the minimum quantity of different results?

How can I solve puzzles like this one? On a different site, two people got two different answers but certainly there can be just one correct.

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    $\begingroup$ Could you please provide proper attribution for this question? For example, a link to that other site. If you want people to comment on another answer, we need to be able to see it. Also, are you asking how to solve the given puzzle, how to solve puzzles from the genre in general, or for us to verify a certain answer? $\endgroup$
    – bobble
    Mar 28 at 4:58

1 Answer 1

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The minimum quantity of different results is

25 different results

This is because

A given integer can be written as a square in at most two ways, x² and (−x)², and as a cube in at most one way. Thus, a given result can be the square or cube of at most three distinct integers. For instance, 64 = 8² = (−8)² = 4³.

To concretely achieve this minimum, we can

Have the initial 75 integers be p², p³, and −p³, for p being each of the first 25 primes. The results will be p⁶ for each of those primes.

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