The Magic of Maths

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First annotation on Sep 7th, 2025. Last on Nov 22nd, 2025.

We call the numbers 1, 3, 6, 10, and 15 triangular numbers, since we can create triangles like the ones below using those quantities of dots. (You might dispute that 1 dot forms a triangle, but nevertheless 1 is considered triangular.) The official definition is that the nth triangular number is 1 + 2 + 3 + · · · + n.

A negative angle moves in the clockwise direction. For example, the angle −30° is the same as the angle 330°. Notice that when you move A degrees in the clockwise direction, you have the same x-coordinate as when you move A degrees in the counterclockwise direction, but the y-coordinates will have opposite signs. In other words, for any angle A, cos(−A) = cos A sin(−A) = − sin A

Proof without words: 1/4 + 1/16 + 1/64 + 1/256 + · · · = 1/3
Here's a little-known fact about this magic square that I call the square-palindromic property. If you treat each row and column as a 3-digit number and take the sum of their squares, you will find that 4922 + 3572 + 8162 = 2942 + 7532 + 6182 4382 + 9512 + 2762 = 8342 + 1592 + 6722 A similar phenomenon occurs with some of the "wrapped" diagonals, too. For instance, 4562 + 3122 + 8972 = 6542 + 2132 + 7982