Count it.....

Probability Level 3

What is the sum of total number of squares and rectangles on a 11 × 11 11 \times 11 chessboard ?

Details

Since rectangles are squares, you are not supposed to double count the squares.


The answer is 4356.

Tushar Malik by Tushar Malik , 20, India

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5 solutions

Saket Sharma
Sep 27, 2014

All sqaures are special case of rectangles (i.e. when the two sides get equal). So, it would be suffice to find number of all the rectangles.

Consider a n x m chessboard. Making a rectangle is equivalent to choosing any 2 points from (n+1) points along n-axis; and choosing any 2 points from (m+1) points along m-axis.

Thus, ( n + 1 2 ) n+1 \choose 2 x ( m + 1 2. ) m+1 \choose 2.

Here we have n = m = 11. So, ( 12 2 ) 12 \choose 2 2 ^{2} = 4356

3 Helpful 0 Interesting 1 Brilliant 0 Confused
Rama Devi
May 21, 2015

The formula for calculating the number of rectangles is 12c2 X 12c2,which results in 4356. Since all the squares can be counted as rectangles,the total number is 4356.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Anandhu Raj
Jan 5, 2015

Number of Rectangles = n c 2 × n c 2 _{ }^{ n }{ { c }_{ 2 } }_\times { }^{ n }{ { c }_{ 2 } }

where n is the number of lines formed by chessboard horizontally and vertically separately.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Alan Naden
Dec 24, 2014

Considering the diagonal of a rectangle it could start in any of the intersections of the lines dividing the board and end at any other so there are (12 * 12) * (11 * 11) = 17424 diagonals. However, note each rectangle has two diagonals each of which can be described in two directions. So 17424 / 4 gives 4356 rectangles.

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Tushar Malik
Sep 8, 2014

All the squares can be counted by the formula n 2 n^{2} + n − 1 2 n-1^{2} ........... 1 2 1^{2} . All the rectangles can be counted by the formula n × n n \times n = [ n ( n − 1 ) / 2 ] 2 [n(n-1)/2]^{2} .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Note that the formula for rectangles should be ( n + 1 2 ) 2 { n+1 \choose 2} ^2 . In particular, I get that the answer should be ( 12 2 ) 2 = 4356 { 12 \choose 2 } ^2 = 4356 .

Can you explain your solution in more detail and provide the steps that you used?

Note that you already said "not supposed to double count the squares", so I don't understand why you're adding n 2 + ( n − 1 ) 2 + … + 1 2 n^2 + ( n-1)^2 + \ldots + 1^2 .

Calvin Lin Staff - 6 years, 9 months ago

Small Typo: "Squares are rectangles", not "rectangles are squares".

Tran Quoc Dat - 5 years, 1 month ago

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