The sum of cubes is just the square of the triangular numbers!
1^{3} + 2^{3}
+ ... + n^{3} =
(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 1^{3} red letter,
2^{3} green letters,
and 3^{3} 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>.
