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Topic: How many rectangles are in an n by n square array of dots?
Replies: 4   Last Post: Sep 9, 2006 11:14 PM

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Oliver Stemforn

Posts: 48
Registered: 9/2/06
How many rectangles are in an n by n square array of dots?
Posted: Sep 8, 2006 9:56 PM
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Let the n by n dots be in one plane, and let n be equal to or greater than 2. Rectangles here are understood to be squares or non-square rectangles. A rectangle counts only when each of its four vertices coincides with one of the dots of the array.***

Can you come up with a polynomial in terms of n for the total number of rectangles for a general n by n array?

*** For instance, if the dots were
(0,0),(0,1),(1,0), and (1,1), (meaning n = 2) then there would be a total of one rectangle for this array.
(In this case it would be described more specifically as a square if connected.)

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