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© 1999-2010 by Francis Edward Su

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?

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.

Su, Francis E., et al. "Sum of Cubes." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Keywords:    proofs without words, number theory
Subjects:    geometry, number theory
Level:    Easy
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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