Sunday, January 23, 2011

\$1.98 Mathematics

A couple years ago, I was walking about a 99-cent store and found a quad-ruled composition notebook and a box of colored pencils. Total: \$1.98. I played around with them for a while and then they were put in a draw and forgotten about. Until recently when I found the notebook.

Here's one of the pictures:

There's a few things going on here. First, there's the demonstrations that two triangular numbers make a square. Second, the overall illustration shows that the sum of consecutive cubes is equal to the square of a triangular number.

Written on paper, there seems to be no reason why:
13 + 23 + 33 + 43 + 53 = 152,
(with 15 being the 5th triangular number)

But the visual shows it to be true. There is one box in the corner, bordering two 2x2 boxes, bordering three 3 x 3 boxes. Granted, I cheated in that all the even numbers contain two rectangles that are 1/2n x n.

I didn't finish coloring it. Probably got bored. Likewise, in the actual notebook, the bottom of the page has four 9 x 9 boxes lightly penciled, but the other five would be off the edge of the paper.

Now that I found the book again, I may start doodling some more... assuming I find the colored pencils.

EDIT: I replaced the image with an annotated version. The original, larger image (click on the picture) is still annotation-free.