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.
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.
