## This One Equation, 10² + 11² + 12² = 13² + 14², Takes Pythagoras To A Whole New Level

“There are 365 days in a (non-leap) year, and 10² + 11² + 12² = 13² + 14² = 365. However, this mathematical fact doesn’t have anything to do with our calendar at all, nor with our planet’s rotation and revolution around the Sun. Instead, the number of days in a year is pure coincidence here, but the mathematical relation is a direct consequence of Pythagorean geometry, something far easier to visualize than just algebra.

Pythagoras just started with a² + b² = c², which has 3, 4, and 5 as the only set of consecutive numbers that solve it. We can extend this as long as we like, however, and for each equation with an odd number of terms we can write down, there’s only one unique solution of consecutive whole numbers. These Pythagorean Runs have a clever mathematical structure governing them, and by understanding how squares work, we can see why they couldn’t possibly behave in any other way.”

You’ve likely seen the Pythagorean Theorem before, which is about the sides and hypotenuses of a right triangle. The only solution of consecutive whole numbers to it is 3² + 4² = 5², which is maybe the simplest right triangle you can imagine. There may not be other consecutive number solutions to this problem, but if you consider having consecutive strings of more numbers, like 5, 7, or any odd number of them, there’s always one unique solution you can find, and they all follow a fascinating pattern.

It’s one of those mathematical instances where sure, you can solve it using algebra, but it’s beautiful and easy to see using geometry. Come enjoy the math of Pythagorean Runs today!