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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?

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  • $\begingroup$ Is $0$ allowed? $\endgroup$ – Lucia Mar 3 '18 at 22:30
  • $\begingroup$ 0 is not allowed $\endgroup$ – Bernardo Recamán Santos Mar 3 '18 at 22:31
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    $\begingroup$ The first problem is surely very easy by the four square theorem. $\endgroup$ – Lucia Mar 3 '18 at 22:32
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    $\begingroup$ 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. $\endgroup$ – Gerhard Paseman Mar 3 '18 at 23:15
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    $\begingroup$ 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. $\endgroup$ – Gerry Myerson Mar 4 '18 at 6:23

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