Mathologer recently published a video on the “Moessner Miracle”, a more advanced version of a known trick in number theory.

As we all learn in elementary or high school, if you take consective odd numbers and add them together, you get all the squares:

     1 2 3 4 5 6 7 8 9 10 . . . .

                1
     1+3     =  4
     1+3+5   =  9
     1+3+5+7 = 16

And so on. What you’re actually doing is skipping every second number. What happens if you skip every third number, add, skip the second number and add again?

     (A skip third )  1 2 3 4  5  6  7  8  9 10 11 12  13 14 15 . . . .
     (B add numbers)  1 3   7 12    19 27    37 48     61 75    . . . . 
     (C skip second)  1 3   7 12    19 27    37 48     61 75    . . . . 
     (D add numbers)  1     8       27       64       125       . . . . 

Congratulations, you’ve produced the cubes. And it turns out you can produce all powers of the first skipped number (x^2 and x^3 in the example cases above). He uses the fifth skip in the video to produce the powers of 5.  He also goes on to show how this also applies to Pascal’s Triangle, combinatorics, directional paths, and other mathematical concepts.

This is way too cool not to share.