# 2601 as Sum of Three Squares in Ten Different Ways

## Theorem

$2601$ can be expressed as the sum of $3$ squares in $10$ different ways.

## Proof

 $\displaystyle 2601$ $=$ $\displaystyle 1^2 + 10^2 + 50^2$ $\displaystyle$ $=$ $\displaystyle 2^2 + 14^2 + 49^2$ $\displaystyle$ $=$ $\displaystyle 10^2 + 10^2 + 49^2$ $\displaystyle$ $=$ $\displaystyle 14^2 + 14^2 + 47^2$ $\displaystyle$ $=$ $\displaystyle 1^2 + 22^2 + 46^2$ $\displaystyle$ $=$ $\displaystyle 14^2 + 17^2 + 46^2$ $\displaystyle$ $=$ $\displaystyle 1^2 + 34^2 + 38^2$ $\displaystyle$ $=$ $\displaystyle 24^2 + 27^2 + 36^2$ $\displaystyle$ $=$ $\displaystyle 17^2 + 34^2 + 34^2$ $\displaystyle$ $=$ $\displaystyle 22^2 + 31^2 + 34^2$

That there are no more can be determined by exhaustion.

$\blacksquare$

## Historical Note

In his Curious and Interesting Numbers, 2nd ed. of $1997$, David Wells reports this result as coming from Volume $22$ of Journal of Recreational Mathematics, page $75$, but this has not been corroborated.