A pictorial proof of the relationship among the sum of cubic integer numbers, sum of consecutive odd numbers, and sum of natural numbers, that is  is provided. By expressing the sum of cubic integer numbers in terms of the number of squares and repeating the rearrangement of squares based on the idea of isoperimetric transformation, it can be understood that the sum of cubic integer numbers can be expressed as the sum of consecutive odd numbers as well as the square of the sum of natural numbers.

sum of cubic numbers, sum of odd numbers, visual thinking.

Received: October 19, 2022; Accepted: November 15, 2022; Published: November 30, 2022

How to cite this article: Yukio Kobayashi, Proof without words: relationship among the sum of cubic integer numbers, sum of consecutive odd numbers, and sum of natural numbers, Far East Journal of Mathematical Education 24 (2023), 1-3. http://dx.doi.org/10.17654/0973563123001

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License 

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