Checkerboard Square Pow

Problem Statement: An 8 by 8 checkerboard is made up of 64 smaller squares. You can combine the squares to form squares of other sizes. How many different squares can be found on a 8 by 8 checkerboard? What if the checkerboard was a different size. Say, 15 by 15. How would you determine the amount of squares on it?

Process: What I did first was I drew an 8 by 8 checkerboard of my own and started counting all of the possibilities.

I then realised that this process would take too long and that I had no patience for this. So I decided to re-read the question and think of another possible way to solve this problem. I thought about what we did in class on thursday and remembered PEMDAS. That was it! I realised that I could try multiplying instead of adding all of the squares. So I made a table counting all of the possible square sizes from 1 by 1 to 7 by 7.

8x8 size

12 =

1

7x7 size

22 =

4

6x6 size

32 =

9

5x5 size

42 =

16

4x4 size

52 =

25

3x3 size

62 =

36

2x2 size

72 =

49

1x1 size

82 =

64

Sum =

204

204

Finally!

This was my answer, 204.

Then I had to find a solution to the second problem. After reading it over carefully, I realised that I had to find an equation. So I asked for some help from my parents and we came up with this:

n(n+1)(2n+1)

Sum = ------------

6

After testing it out, the equation was correct.

Solution: I came up with 204 possible squares that can be found on an 8 by

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