A Friday in April/March 2014, I can’t remember the exact date.

I was walking home with Demetrius, taking the usual path that people use when they actually want to go to school (to some extent).

I was to D about my Graphics coursework, holding the chess board I was going to take home to test on.

We bumped into Mr Okyere on the way home. He wanted to know what my project was, and so I told him.

He then challenged me to figure out how many squares there are on an 8x8 grid.

Philosophically speaking, you’d be automatically wrong for thinking 64.

On the following Monday, Mr Okyere showed me how to go about it:

You use square numbers.

There is 1 8x8 grid, that’s a total of 64 squares.

Next, how many 7x7 grids are there?

4. There are 4 of them. That’s 4 x 49, which is 196.

9 6x6 squares are next. That’s 324 individual squares.

Next, there are 16 5x5s, that’s 400.

25 4x4s, that’s 25 x 16, or 400, again.

36 3x3s, 324 squares.

49 2x2s, 196 squares.

And lastly, there’s 64 1x1 squares – so that’s 64.

But the answer is not 1968 – the total of individual squares from the 8 combinations. The answer actually lies in adding up the square numbers:

**64** 1x1s

**49** 2x2s

**36** 3x3s

**25** 4x4s

**16** 5x5s

**9** 6x6s

**4** 7x7s

And **1** 8x8.

That’s 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1, that’s 204 in total.

That’s what Mr Okyere had taught me. Then I did my own research.

It turns out that 204 is special in that:

It’s a pyramid square number – a pyramid square number is a number that represents the number of stacked spheres in a pyramid in a square base. 204 is the 8^{th} pyramid square number.

It’s a nonagonal number. Like square numbers – the number of perfect squares you can make with dots, or really anything for that matter – you can have nonagonal numbers, 204 is the 8^{th}. Again.

It’s a truncated triangular pyramid number. Something to do with triangular pyramid numbers and frustums, I think.

It’s the number for the H.T.T.P. code saying your request was received but there was no response.

It’s the exact number that represents the number of possible ways to place 3 queens on a 5x5 chess board – 3 queens who aren’t attacking.

It’s the number that represents how many hands you can have in a poker deck with 1 wild joker that’s at least as good as a straight flush. In poker, a straight flush is when you have a hand of cards that are all in 1 suit and all in a continuous sequence (e.g.: ace of hearts, 2 hearts, 3 hearts, 4 hearts, 5 hearts).

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