Sum of Cubes

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

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. <http://www.math.hmc.edu/funfacts>.

References: John H. Conway and Richard K. Guy, The Book of Numbers, pp.58. Keywords:    proofs without words, number theory Subjects:    geometry, number theory Level:    Easy Fun Fact suggested by:   Francis Su Suggestions? Use this form. 4.20 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts! Get the Math Fun Facts iPhone App!

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