Math!

whileyouweresleeping:

Steven Strogatz, in Opinionator:

… suppose we add all the consecutive odd numbers, starting from 1:

1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

The sums above, remarkably, always turn out to be perfect squares. (We saw 4 and 9 in the square patterns discussed earlier, and 16 = 4 × 4, and 25 = 5 × 5.) A quick check shows that this rule keeps working for larger and larger odd numbers; it apparently holds all the way out to infinity.

Did that just blow your mind, too?

— From London.

Neat, huh? But there’s a pretty intuitive explanation: suppose you have a bunch of pennies arranged in a square, and you want to increase the dimensions of the square by one. To do so, you have to add an L-shaped collection of pennies, and the L-shape always has an odd number of pennies in it.

The Wikipedia page on square numbers illustrates this perfectly:

Count the pink diamonds in each figure: 1, 3, 5, 7, 9…