Sum of Cubes

The sum of cubes is just the square of the triangular numbers!

13 + 23 + … + n3 = (1 + 2 + … + n)2.

And there is a nice proof by picture, too. Can you figure out how this diagram illustrates the identity?

ABBCCC
BAABBB
BAABBB
BCCAAA
BCCAAA
BCCAAA

Presentation Suggestions:
Draw this picture and see if your students can figure out why the diagram is a “proof without words”!

The Math Behind the Fact:
The diagram illustrates the identity for n=3. Note that the square has (1+2+3)2 letters in it. But now I also claim that there is 13 red letter, 23 green letters, and 33 blue letters.

This can be seen by arranging the letters in “layers” of a cube! The red cube has one layer (A). The green cube has two layers (A and B) with 4 letters in each. The blue cube has three layers (A, B, and C) with 9 letters in each.

This construction easily generalizes for arbitrary n. You can follow this with the Fun Fact Sum of Cubes and Beyond.

How to Cite this Page: 
Su, Francis E., et al. “Sum of Cubes.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
John H. Conway and Richard K. Guy, The Book of Numbers, pp.58.

Fun Fact suggested by:
Francis Su

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