Ugh question about checkerboard squares....?

Question

Okay so we know that there's 204 squares on a checker-board but what if we have a checkerboard of a different size?

What if we have a checkerboard of a different size, not 8x8. How can we determine how many squares are on it altogether?

Answer 1

With a 1 x 1 checkerboard, you can form 1 square = 1

With a 2 x 2 checkerboard, you have the big square, plus 4 little squares = 5

With a 3 x 3 checkerboard, you have the big square, plus 4 squares of size 2x2 and the 9 little squares = 14

etc.

Basically, if you have a n-sided checkerboard the answer is:

1 + 4 + 9 + 16 + ... + n²

The shortcut formula for that is:

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

As a check, try some numbers:

n = 1

1(2)(3) / 6 = 6/6 = 1

n = 2

2(3)(5) / 6 = 30/6 = 5

n = 3

3(4)(7) / 6 = 84/6 = 14

n = 4

4(5)(9) / 6 = 180/6 = 30

...

n = 8

8(9)(17) / 6 = 1224/6 = 204

Now you try it with a different value of n...

Answer 2

8 x 8 = 64

204/64 = 3.1875

If the squares are the same size, but on a different sized board you can multiply the dimensions together, then multiply that answer by 3.1875 to get the approximate number of squares.

EX: 11 x 11 board

11 x 11 = 121

121 x 3.1875 = 385.6875 ~ 386 squares