How many squares are there all in all in a chessboard?

Question

I don't only mean the squares you could see plainly. There are 64 squares. What I need is them total # of squares in all. If you put 4 small squares together, it will make one big square, and so on and so forth. Pls. help.

Answer 1

Consider the lefthand vertical edge of a square of size 1 x 1.
This edge can be in any one of 8 positions. Similarly, the top
edge can occupy any one of 8 positions for a 1 x 1 square.
So the total number of 1 x 1 squares = 8 x 8 = 64.

For a 2 x 2 square the lefthand edge can occupy 7 positions and
the top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2.

Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

We can summarize the results as follows:

Size Of square Number of squares
--------------- -----------------
1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204

Answer 2

Well, let's do this systematically.

First you have the 64 basic squares.

Then you can take 4 squares and declare this to be a bigger square.
You can have 7 of those in a row, and 7 such rows. So that is 49 different 2 by 2 quares.

Then at 3 by 3, you can have 6 by 6.

4 by 4, there will be 5 times 5.

and so on.

So in summary:

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

for a total of 204

Answer 3

Consider the lefthand vertical edge of a square of size 1 x 1.
This edge can be in any one of 8 positions. Similarly, the top
edge can occupy any one of 8 positions for a 1 x 1 square.
So the total number of 1 x 1 squares = 8 x 8 = 64.

For a 2 x 2 square the lefthand edge can occupy 7 positions and
the top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2.

Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

We can summarize the results as follows:

Size Of square Number of squares
--------------- -----------------
1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204

Source(s):

dr math


********make that 205 including dr math ;-)

Answer 4

I don't suppose that anyone mentioned the reason for the trend. If you notice that the number of squares on an NxN board is:

N^2 + (N-1)^2 + ...+ (2)^2 + (1)^2

The the series is the sum of the first N perfect squares.

I.e. for a 4x4 board, the answer would be:
4^2 + 3^2 + 2^2 + 1^2 = 30

Answer 5

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares

204 squares

Answer 6

64

Answer 7

do you mean just squares that are made up of full squares with the corners where the corners of the original squares are?

or can we have squares at an angle with the corners on the original squares

or can the corners start any old where at any angle?

the problem with this question is that more specific information is required

Answer 8

It obviously isn't 204 if u are considering rectangles as well. With only squares, the answer is clearly 204, but when considering rectangles as well, the number is actually 1296.

Answer 9

204

Answer 10

204