Amazingly, the following mathematical identity is true:
3 3 3 3 2 1 +2 +3 +...+n = (1+2+3+...+n)
Thus, for example,
1 = 1 1 + 8 = 3 * 3 1 + 8 + 27 = 6 * 6 1 + 8 + 27 + 64 = 10 * 10
It would seem that to prove this requires a lot of complicated and messy algebra. But, actually, it's possible to see right away why it is true...
by looking at the diagram above.
I first encountered it in Martin Gardner's Mathematical Games column in the October, 1973 (an ill-omened month indeed!) issue of Scientific American wherein he credits Dr. Solomon W. Golomb for originating it, it having first appeared in the May, 1965 issue of The Mathematical Gazette.