# 2601 as Sum of 3 Squares in 12 Different Ways

## Theorem

$2601$ can be expressed as the sum of $3$ squares in $12$ different ways.

## Proof

 $\displaystyle 2601$ $=$ $\displaystyle 51^2$ $\displaystyle$ $=$ $\displaystyle 1^2 + 10^2 + 50^2$ $\displaystyle$ $=$ $\displaystyle 2^2 + 14^2 + 49^2$ $\displaystyle$ $=$ $\displaystyle 10^2 + 10^2 + 49^2$ $\displaystyle$ $=$ $\displaystyle 14^2 + 14^2 + 47^2$ $\displaystyle$ $=$ $\displaystyle 1^2 + 22^2 + 46^2$ $\displaystyle$ $=$ $\displaystyle 14^2 + 17^2 + 46^2$ $\displaystyle$ $=$ $\displaystyle 1^2 + 34^2 + 38^2$ $\displaystyle$ $=$ $\displaystyle 14^2 + 31^2 + 38^2$ $\displaystyle$ $=$ $\displaystyle 3^2 + 36^2 + 36^2$ $\displaystyle$ $=$ $\displaystyle 24^2 + 27^2 + 36^2$ $\displaystyle$ $=$ $\displaystyle 17^2 + 34^2 + 34^2$ $\displaystyle$ $=$ $\displaystyle 22^2 + 31^2 + 34^2$

That there are no more can be determined by exhaustion.

$\blacksquare$

## Sources

• 1989: John M. HowellProblems and Conjectures: $\text Q 1692$. Three Squares (J. Recr. Math. Vol. 21, no. 1: p. 68)
• 1990: Solutions to Problems and Conjectures: $\text Q 1692$. Three Squares (J. Recr. Math. Vol. 22, no. 1: pp. 74 – 76)