How many squares of any size are there on an 8x8 chessboard?

Question

A. 4096
B. 1296
C. 204
D. 65
E. 64

Kvadlamani7 · Accepted Answer

Hi,

Think the question asks for any size so 4 small squares of 1*1 and 1 large square of 2*2 making it 5 squares in total. The easiest way to solve this que would be to n^2+(n-1)^2+...(n-n)^2.

On those lines the answer should be 8^2+7^2+6^2+.....1^2 = 204.

parth2424 · Answer

If 2x2 chess board - literally draw it squares are 4 
As 2 rows and 2 columns so 8x8 
64 squares

Posted from my mobile device

kjchiang3 · Answer

.1..2...3...4..5...6..7...8
                1
                2
                3
                4
                5
                6
                7
                8

The question is asking how many different squares of differing sizes you can make out of a chessboard.
For example, you can make 64 different 1x1 size squares, as there are 64 different squares.
Alternatively, you can make 1 single 8x8 size squares, as that is as large as the chessboard is.

e.g. For a 5x5 size square, how many different ways can you position it?

.1..2...3...4..5...6..7...8
                  1
                  2
                  3
                  4
                  5
                6
                7
                8

In this example for a 5x5 square, you can see that there are 4 positions it can take vertically and horizontally.
The current position, and 3 others vertically and horizontally. 
So, there are a total of 4x4 ways to position a 5x5 square.

Now to find how many 2x2, 3x3, 4x4, 5x5, 6x6, and 7x7 squares with different positioning on the chessboard.

Extrapolating this...
Size  .....  Possible Squares
 1 ............ 64 (8x8)
 2 ............ 49 (7x7)
 3 ............ 36 (6x6)
 4 ............ 25 (5x5)
 5 ............ 16 (4x4)
 6 ............ 9 (4x4)
 7 ............ 4 (4x4)
 8 ............ 1 (1x1)

Sum of which is 204.

KarishmaB · Answer

You need squares of any size. Imagine an 8 inches by 8 inches chessboard with 64 squares of 1 inch by 1 inch. Now does the chessboard have only 64 squares? It has 64 squares of size 1x1 but what about squares of size 2x2? or 3x3 etc? If you take 4 small squares together (like the orange square at the top left corner), you get squares of size 2x2. Move one step to the right and you get another such square and keep going till you get the 7th square at the end. Similarly if you go down one step, you get another square till you get 7 squares. So overall, you will get 7 * 7 = 49 squares of size 2x2. 2-by-21.jpg Similarly you will get squares of size 3x3, size 4x4 and so on till you get one big square of size 8x8. Total number of squares = 8*8 + 7*7 + 6*6 + 5*5 ... 1*1 = 8^2 + 7^2 + 6^2 + ... + 2^2 + 1^2 If you know the formula, use it: 1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1)/6 When n = 8, Sum = 8*9*17/6 = 204 Else you can manually calculate.

KevoK · Answer

My approach to this: I just draw out an 8x8 grid. Then, for every size of the square, I just count it, marking it using with a curve from the 1st to the last square on the axis :

1x1 : 64 (this is obvious)
2x2 : 49 (7 on the horizontal axis, and it should be the same on the vertical axis, thus 7*7)
3x3 : 36 (same as above)

from here, I noticed the pattern and just calculated 8*8 + 7*7 + 6*6 +.... 1*1

Thus C. 204

How many squares of any size are there on an 8x8 chessboard?

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How many squares of any size are there on an 8x8 chessboard?

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