How many total squares are there in a checkerboard/chessboard...?

Question

...including every sized square? That means 1by1 squares, 2by2 squares, 3by3 squares. I know there is not 64 squares since that would only account for all the one by one squares. I also ask for a formula that will help you find the number of total squares for any n x n chessboard.

Answer 1

Start by taking a look at all the total of each size square.

1 x 1 = 64
2 x 2 = 49
3 x 3 = 36
4 x 4 = 25
5 x 5 = 16
6 x 6 = 9
7 x 7 = 4
8 x 8 = 1

Adding them all = 204

Notice that is the sum of n^2 and that formula is
sum=(1/6)n(n + 1)(2n + 1)

Answer 2

There are 204 squares (of all sizes; it's the total) on a chessboard.

Think of a chessboard as a set of 2 types of positions, vertical and horizontal. Now, the square types vary in dimension (there's 1x1, 2x2, 3x3...8x8).

Consider the number of 1x1 squares:

There are 8 locations where these squares can be located horizontally, and 8 where they can be located vertically. So the total number of 1x1 squares is 8x8=64 (which, of course, you already know).

Now let's move on to 2x2. When we form a 2x2 square, we use up 2 squares from both the horizontal "location" and the vertical one (as opposed to using 1 from each), so the number of spaces it can occupy decreases by 1. So there are 7 locations horizontally and 7 vertically, so 7x7=49 2x2 squares.

If you continue this process, you will see that the pattern is the same. For 3x3, the number of squares is 6x6=36, for 4x4 is is 5x5=25 squares, for 5x5 is is 4x4=16, then for 6x6 we get 9, for 7x7 we get 4, and then there is the 1 big square of 8x8.

Now we just add:
64+49+36+25+16+8+4+1=204 squares.

So now you want a formula for the total number of squares on a nxn board.

Think back to how we solved this:

204=8x8+7x7+6x6+5x5+4x4+3x3+2x2+1x1

So for an nxn board, the formula is:

nxn+(n-1)x(n-1)+(n-2)x(n-2)+...+1x1

=[n(n+1)(2n+1)]/6

Answer 3

For an 8x8 board, there are...
1 square that is 8x8
4 squares that are 7x7
9 squares that are 6x6
16 squares that are 5x5
25 squares that are 4x4
36 squares that are 3x3
49 squares that are 2x2
64 squares that are 1x1

So, the formula for the number of squares on an n x n board is:

RiemannSum[i^2], from i=1 to i=n
which is equivalent to
(1/6)*n*(n+1)*(2*n+1)

That is, simply sum the first n squares. A 8x8 board would yield

1+4+9+16+25+36+49+64 = 204 squares

Answer 4

8 by 8 board has 204 squares.

Sigma (n^2), n=1 to n

Answer 5

n(n+1)(2n+1)/6 = 8(9)(17)/6 = 204

Answer 6

92 including the board itself.