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 Squares and Rectangles (Posted on 2004-01-21)
There is a grid of 20 squares by 10 squares. How many different rectangles are possible?
(Note that square is a rectangle).

 See The Solution Submitted by Ravi Raja Rating: 2.0000 (5 votes)

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 One direction at a time | Comment 14 of 20 |
Its quite easy to see this one is you consider one direction at a time. Take a general rectangle of a x b squares and consider first the "a" direction.

There is first of all the largest possible side (a)

Then there are 2 possible ways of placing the length (a-1), 3 ways of placing (a-2) etc all the way down to the single square where there are "a" ways. Adding up there is the sum:
1+2+3+4+5+6+......+a
The sum of an arithmetic series is the average of the first and last term multiplied by the number of terms i.e. (1+a)*a/2

Similarly in the b direction there are (1+b)*b/2 possibilities. So overall in both directions there are (1+a)*a*(1+b)*b/4.

The values here give 11,550.

Interestingly this can be taken to three dimensions - (1+a)*a*(1+b)*b/6 etc

 Posted by Jils on 2004-01-22 12:05:49

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