How many squares does a chess-board contain?

Question

Answer with proper reasoning.

Hint: The answer isn't 64.

Answer 1

formula  = n(n+1)(2n+1)/6

n - no of rows

Total = 204 + 1(reverse side of board).. = 205 Squares.

Answer 2

208

Answer 3

16 perfect squares. You don't count each individual square. You group the other squares together to make 16

Answer 4

64 i think

Answer 5

8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 +1 =204

Can't think of a formula though

Answer 6

As the title suggests, the investigation involves children finding out how squares there are on a chessboard. You might think that there are only 64, but you would be wrong...

The diagram below shows that there are indeed 64 squares, you there are also some more...

And, also the 16 two-by-two squares shown below (although these aren't the only 2x2 squares!)...

There are many more different-sized squares on the chessboard.

The complete list of answers is shown below:

1, 8x8 square

4, 7x7 squares

9, 6x6 squares

16, 5x5 squares

25, 4x4 squares

36, 3x3 squares

49, 2x2 squares

64, 1x1 squares

Therefore, there are actually 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 squares on a chessboard! (in total 204).

A worksheet with a large chessboard which children can use to investigate this problem can be found here.

If the children manage to find all of them, ask them if they can see a pattern in the results (i.e. the square numbers in the table)

Answer 7

1×1 squares: 8×8 = 64

2×1 squares: 7×7 = 49

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8×8 squares: 1×1 = 1

Total number of squares = 8²+7²+...+1² = 204

1² + 2² + 3² + ....n² = n(n+1)(2n+1)/6

You can prove this formula to be accurate using mathematical induction

Answer 8

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

Answer 9

64+4+16+1=85.

Answer 10

8x8=64 64 1x1 squares 49 2x2 36 3x3 25 4x4 16 5x5 9 6x6 4 7x7 1 8x8 204 squares all together!

Answer

8x8=64

64 1x1 squares

49 2x2

36 3x3

25 4x4

16 5x5

9 6x6

4 7x7

1 8x8

==

204 squares all together!

There are 64 squares on a standard chess board.

Interestingly enough, if you put one grain of rice on one square, two on the next, four on the next, eight on the next, and continued doubling the number on each square, there aren't enough grains of rice in the world to finsh. (notwithstanding the room they'd need...)

The chess board has 64 squares, alternating between the two most popular colors of black and white.

  

Answer 11

there are 64 simple squares

there are also squares of 2 by 2, etc... till 8*8

to find how many 2*2 squares there are we can look at the placement of the squares top left part and it cannot be on bottom row or right column

so there are 7*7 such squares

the complete formula is the sum for i=1..8 of (8-i+1)^2=

8*8+7*7+...2*2+1=204