# Checkerboard puzzle

Take a look at the checkerboard below. A typical checkerboard puzzle could ask you the following question:

How many squares, of all sizes, are there on this 8 × 8 checkerboard? However, isn't it important to know how many different sizes are there?

The different sizes are 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8 In this checkerboard puzzle, it is easy to know how many 1 × 1 there are. Since there are 8 such square on each size, there are a total of 8 × 8 = 64

It is also easy to see that there is only 1 square that has a size of 8 × 8

To find out how many there are for any other size is a big headache. However, I will illustrate the technique or trick for 2 sizes, the 2 × 2 and the 6 × 6.

I leave it up to you to find it for the remaining sides! And if you do find it, I am happy.

First, let us find out how many 2 × 2 there are. You will need to carefully examine the following illustration: I carefully numbered all the 2 × 2 squares we can get on one side starting from the very top and going down 1 unit each time

Since there are 7 such square on one side, we know there will be 7 such square on any other side.

since 7 × 7 = 49, 49 squares have a size of 2 × 2

Next, let us find out how many 6 × 6 there are. Again, you will need to carefully examine the following illustration: I carefully numbered all the 6 × 6 squares we can get starting from the very top and going down 1 unit each time

Since there are 3 such square on one side, we know there will be 3 such square on any other side.

since 3 × 3 = 9, 9 squares have a size of 6 × 6

Following a similar course, there are

36 squares with a size of 3 × 3

25 squares with a size of 4 × 4

16 squares with a size of 5 × 5

4 squares with a size of 7 × 7

Adding the values of all sizes, we get 64 + 1 + 49 + 9 + 36 + 25 + 16 + 4 = 204

Therefore, there are 204 squares of sizes 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8

Have fun with this checkerboard puzzle! Any questions? contact me using the form in about me

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!