Question 196
We have $n \geq 3$ sheets numbered from 1 to $n$. The sheets are then divided into 2 piles and we determine whether at least 1 pile contains sheets marked with numbers whose sum is a perfect square. Prove that
a) If $n \geq 15$, such sheets exist regardless of how the sheets are split.
b) If $n \leq 14$, no such sheets exist in every distribution.
a) If $n \geq 15$, such sheets exist regardless of how the sheets are split.
b) If $n \leq 14$, no such sheets exist in every distribution.