Blog 3: How Many Squares Are in A Chessboard?

For this fun activity, we are given an 8x8 chessboard. How many squares are there?

Before we get excited and jumping into conclusions, if you are a visual person like me, you can start by either having the chessboard in front of you, or you can start by drawing an 8x8 grid on a piece of paper.

1. Once you have the visual in front of you, let's start by looking at 1x1 grid.

Think:
How many 1x1 grids are in an 8x8 chessboard?

Visual example.

Answer:
Since there are 8 pieces of 1x1 grid horizontally and 8 pieces of 1x1 grid vertically, we can find the total numbers of 1x1 grids by calculating the area.

There are 64 pieces in total. (You can count them all to be sure!)

2. We now look at 2x2 grid.

Think:
How many 2x2 grids are in an 8x8 chessboard?
Hint: think about how many 2x2 grids are there horizontally and vertically. Include any (overlapping) combinations of 2x2 squares, as seen in red.

Visual example.

Answer:
If you continue to find 2x2 grids, we can see that there are 7 pieces of 2x2 grid horizontally and 7 pieces of 2x2 grid vertically.
There are 49 pieces of 2x2 in total.

3. We now look at 3x3 grid.

Think:
How many 3x3 grids are in an 8x8 chessboard?
Hint: think about how many 3x3 grids are there horizontally and vertically. Again, include any (overlapping) combinations of 3x3 square, as seen in red.

Visual example.

Answer: 
We can see that there are 6 pieces of 3x3 grid horizontally and 6 pieces of 3x3 grid vertically.
There are 36 pieces of 3x3 in total. 

Do you notice any patterns?

See how as the grid size increases by 1, the numbers of pieces of square decreases by 1? As the pattern continues, we can create a handy-dandy table that tracks how many squares are in each case. You do the calculations!

So now that you have counted or calculated how many squares there are from 1x1 grid to 8x8 grid, what's next?

We can add the numbers of grids of all sizes together!

Why?

Let's look at a similar (simple) example: How many squares are in a 2x2 chessboard?

A 2x2 chessboard is consists of 4 pieces of 1x1 grid and 1 piece of 2x2, since the chessboard itself is a square. In total, there are 4+1=5 squares.

Final answer:
Similarly, in an 8x8 chessboard example, we have to add all the numbers of squares together because we are looking for the total number of squares.

In total,
There are 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares in an 8x8 chessboard.

Check: Did you do the calculations right?


This puzzle can be extended if we ask how many rectangles of a certain unit sizes are in the 8x8 chessboard. For instance, we might ask: Find the numbers of 2x3 rectangles are in the chessboard, or find the numbers of 3x2 rectangles in the chessboard. To make this activity even more interesting, we can extend this chessboard to a 3-dimensional objects, like a Rubik's cube! This might pose a challenge to some students, but it is certainly more fun as we get our brains moving!