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.

A 4 by 4 grid drawn in blue pen. Two 3 by 3 squares are shaded in with pencil. One of these squares covers the bottom left of the grid. The other square covers the top right. The two squares overlap in the centre of the grid.

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:

A 4 by 4 grid drawn in blue pen. Two 3 by 3 squares are shaded in with pencil. One of these squares covers the bottom left of the grid. The other square covers the top right. The two squares overlap in the centre of the grid.

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.

A 4 by 4 grid drawn in blue pen. Three 2 by 2 squares are drawn across the top half of the grid and labelled 1, 2, and 3. The first box covers the top left quadrant of the grid. The third box covers the top right quadrant of the grid. The middle box is in the centre overlapping the other two solutions.

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 sizeNumber of squares
4 by 41
3 by 34
2 by 29
1 by 116

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 = 12, 4 = 22, 9 = 32, 16 = 42). 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.

One response

  1. Paul Avatar

    And there is a formula for finding the sum of consecutive squares up to n-squared.

    Sum = n(n+1)(2n+1)/6.

    In the case of the chessboard, sum = 8x9x17/6 = 204.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

By submitting this form, you give CSIRO permission to publish your comments on our websites. Please make sure the comments are your own. For more information please see our terms and conditions.

Why choose the Double Helix magazine for your students?

Perfect for ages 8 – 14

Developed by experienced editors

Engaging and motivating

*84% of readers are more interested in science

Engaging students voice