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From the Fun Fact files, here is a Fun Fact at the Easy level:

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?


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. <>.

Keywords:    proofs without words, number theory
Subjects:    geometry, number theory
Level:    Easy
Fun Fact suggested by:   Francis Su
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