MathDB

Let

n

be a positive integer. Mary writes the

n^3

triples of not necessarily distinct integers, each between

1

and

n

inclusive on a board. Afterwards, she finds the greatest (possibly more than one), and erases the rest. For example, in the triple

(1, 3, 4)

she erases the numbers 1 and 3, and in the triple

(1, 2, 2)

she erases only the number 1,
Show after finishing this process, the amount of remaining numbers on the board cannot be a perfect square.

Problem Statement