Chess Board

Probability Level 2

In permutations and combinations , one of the problems that appear frequently is asking for the total number of squares on 8 x 8 chessboards. Can you tell what the answer is?

As an extension, can you find a formula to count the number of squares in N x M rectangle ?

205 204 102 64

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Prashant Mudgal by Prashant Mudgal , 34, India

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4 solutions

Sandeep Bhardwaj
Aug 26, 2014

For any n × m n \times m rectangle

No. of squares = n . m + ( n − 1 ) ( m − 1 ) + ( n − 2 ) ( m − 2 ) + . . . . . . . . . . . . . . . . . = n.m + (n-1)(m-1)+(n-2)(m-2)+................. upto when any one of the two terms in multiplication becomes zero

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you can also use 8^2+(8-1)^2+(8-2)^2+.....+(8-8)^2

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

=204

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Palash Som
Oct 5, 2014

APPLY THE FORMULA N (N+1) (2N+1)

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Ok I get the answer. But I hv a doubt by ths do u mean tht there are 204 squares ina 8*8 chess board ?

shobitha gunashekaran - 6 years, 4 months ago

Then we'll get 1224.

The formula is n(n+1)(2n+1)/6.

Sai Ram - 6 years ago

Damn wrong! WHO WILL DIVIDE BY SIX?

Agastya Chandrakant - 6 years, 4 months ago

Read this: http://puzzles.nigelcoldwell.co.uk/twentyseven.htm

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