# Squares as sum of squares

For which positive integers n is $n^2$ the sum of precisely n smaller positive squares?
Of these n x n squares, which can be actually cut into n smaller squares?

• Is $0$ allowed? – Lucia Mar 3 '18 at 22:30
• 0 is not allowed – Bernardo Recamán Santos Mar 3 '18 at 22:31
• The first problem is surely very easy by the four square theorem. – Lucia Mar 3 '18 at 22:32
• For even numbers larger than 2, one can make one of the geometric squares have side n-2. Gerhard "Still Working On Odd Case" Paseman, 2018.03.03. – Gerhard Paseman Mar 3 '18 at 23:15
• If $n$ odd is large enough, I'm guessing one can use a square of side $n-3$, together with an appropriate number of squares of sides $1$, $2$, and $3$. E.g., $n=99$, take 59 of side 3, 5 of side 2, and 34 of side 1. – Gerry Myerson Mar 4 '18 at 6:23