Pentagonal number theorem

Imagine you have a bunch of blocks that you want to stack in the shape of a pentagon. Each row of blocks has one more block than the row above it. The first row has one block, the second row has two blocks, and so on.

The number of blocks you used to build the pentagon is called a pentagonal number. The formula to find out how many blocks you need for a pentagonal number is:

n(3n-1)/2

Now, the pentagonal number theorem tells us that any natural number can be expressed as the sum of three pentagonal numbers. This means that you can take any number and break it down into three pentagonal numbers. It's like taking a big puzzle and breaking it down into smaller pieces.

For example, let's say you have the number 15. You can break it down into three pentagonal numbers like this:

15 = 12 + 2 + 1

Where 12 is the 4th pentagonal number, 2 is the 2nd pentagonal number, and 1 is the 1st pentagonal number.

This theorem was discovered by the mathematician Euler in the 18th century and it has been used in many areas of math and science since then. So, remember, if you have a number, you can break it down into three pentagonal numbers thanks to the pentagonal number theorem!