**Difficulty: Extreme**

Rumi really likes chess. One day, her chess club coach asks her, “how many squares are on a chessboard?”

Rumi replies, “That’s easy, the board is 8 by 8, so there are 64 squares!”

Her coach smiles, “That’s true but what about the larger squares made up of smaller squares? They could be 2 by 2 squares, 3 by 3 squares and so on. The board is an 8 by 8 square after all!”

**Warm up question: ** How many squares are on a 4 by 4 board?

**Challenge question: **How many squares are on a full 8 by 8 chessboard?

## Need a hint?

We recommend drawing a 4 by 4 grid in pen and then shading in different squares with a pencil. This way you can explore and erase and explore some more!

We also recommend starting from the biggest square and working your way down. There is one 4 by 4 square, but how many different 3 by 3 squares fit in the grid? Remember that the solutions can overlap with each other.

Once you have found the total for each type of square, do you notice a pattern? What do these numbers have in common?

## Brainteaser answer

Rumi, after some thought and a bit of maths, will tell her chess club coach that a 4 by 4 board has 30 squares and a full 8 by 8 chess board has 204 squares.

Let’s start with the 4 by 4 board and look for patterns we can apply to the larger chess board.

First, we drew a 4 by 4 board in pen so we could pencil in squares, erase, and explore again! We started with the largest square size and worked our way down. There is only one 4 by 4 square, the board itself.

Counting the 3 by 3 squares is a little harder because there are several ways a 3 by 3 square can fit inside the board. Remember the solutions can overlap with each other. For example, here are two different solutions:

Working from our picture, you’ll notice that each square goes into one corner of the board. Since there are four corners of the board, there are four 3 by 3 squares on the board.

There are more ways to fit a 2 by 2 square in the grid, so let’s be systematic. Start by asking: how many squares can we put along the top of the board? We could put one in each corner, and there’s a third we can put in the middle that doesn’t touch the corners at all.

Next, we asked: how many 2 by 2 squares fit along the left side of the board? Since the board is a square, each of its sides is the same, so you won’t be surprised it also can fit 3 squares.

Here’s the tricky step. You have a column of 2 by 2 squares running down from the top left square. You can draw a similar column of squares under the top middle square, and under the top right square too!

Calculating it out, you have three columns of three squares, for a total of nine 2 by 2 squares.

Finally, finding the number of 1 by 1 squares is found by multiplying the length and width of the grid, or 4 times 4, which comes to 16.

Let’s summarise our results and look for a pattern:

Square size | Number of squares |

4 by 4 | 1 |

3 by 3 | 4 |

2 by 2 | 9 |

1 by 1 | 16 |

TOTAL | 30 |

We have our first answer: there are 30 total squares in a 4 by 4 grid.

You might also notice that all of the subtotals are perfect squares (1 = 1^{2}, 4 = 2^{2}, 9 = 3^{2}, 16 = 4^{2}). Why? Recall how we calculated the number of 2 by 2 squares. The number of squares that fit along the length and the width of a square grid will always be the same. And multiplying two of the same number will always give us a square number.

Let’s apply this pattern to the chess board. We know there is only one 8 by 8 board, and we also know there are 64 of the smallest 1 by 1 squares. To find the total number of squares we need to add up all the perfect square numbers between 1 and 64. That’s 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204.

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