number of rectangles in a chessboard...

[psuresh1982]

If a chessboard contains 7 rows and 7 columns, how many rectangles are there be in total ? you include squares too beause square is a special kind of rectangle.

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suresh

[jamesravid]

Answer is 28^2 = 784.

1^3 +2^3 +...+7^3 = (n(n+1)/2)^2

[psuresh1982]

Yes james...You are absolutely correct...Here is my detailed solution for this puzzle...


The following statement shows the number of possibilities for different lengths of the rectangles on a 7 x 7 board:

Length of rectangle Number of Possibilities
7 units 1
6 units 2
5 units 3
... ...
1 unit 7

So, number of possibilities for different lengths of rectangles = 1 + 2 + 3 + ... + 7 = 28.
Similarly, number of possibilities for different breadths of rectangles = 1 + 2 + 3 + ... + 7 = 28.
Hence, number of rectangles = 28 x 28 = 784.

Another way:

Is there a formula for the sum of the first n positive integers ?
Is 1 + 2 + 3 + 4 + ... + n = n (n + 1) / 2 ?

Can this puzzle be solved quickly with knowledge of permutations and combinations?
Note nC2 is the number of combinations of n things taken 2 at a time.
nC2 = n (n - 1)/2. Hence the number of rectangles = 8C2 x 8C2 = 28 x 28 = 784.


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suresh