PROOF WITHOUT WORDS: SUMS OF SUMS OF SQUARES

Abstract

A pictorial proof of the Sums of Sums of Squares, that is,

$$
4\left(1^2+\left(1^2+2^2\right)+\left(1^2+2^2+3^2\right) \cdots\left(1^2+\cdots+n^2\right)\right)=2(n+1) \sum_{k=1}^n T_k
$$

(T_k : Triangular Number), is provided. By expressing each square number as a sum of consecutive odd numbers, we can place those odd numbers into a tetrahedron. And if we change the base of this tetrahedron four times, we get four tetrahedrons. By summing up the corresponding parts of these four tetrahedrons, the Sums of Sums of Squares can be expressed as the formula shown above.

Received: December 2, 2022 
Accepted: December 10, 2022