This simple multiplication table shows the first 20 perfect squares along the diagonal of the table. Bizarrely enough, not only does 3² + 4² = 5², but 10² + 11² + 12² = 13² + 14². There is more to this relation than mere coincidence. (PUBLIC DOMAIN)

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

Unbelievably enough, it all comes back to Pythagoras.

One of the first theorems anyone learns in mathematics is the Pythagorean Theorem: if you have a right triangle, then the square of the longest side (the hypotenuse) will always equal the sums of the squares of the other two sides. The first integer combination that this works for is a triangle with sides 3, 4, and 5: ³² + ⁴² = ⁵². There are other combinations of numbers that this works for, too, including:

• 5, 12, and 13,
• 6, 8, and 10,
• 7, 24 and 25,

and infinitely more. But 3, 4, and 5 are special: they’re the only consecutive whole numbers that obey the Pythagorean Theorem. In fact, they’re the only consecutive whole numbers that allow you to solve the equation a² + b² = c² at all. But if you allowed yourself the freedom to include more numbers, you could imagine that there might be consecutive whole numbers that worked for a more complex equation, like a² + b² + c² = d² + e². Remarkably, there’s one and only one solution: 10² + 11² + 12² = 13² + 14². Here’s why.