
How many rectangles are in an n by n square array of dots?
Posted:
Sep 8, 2006 9:56 PM


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 nonsquare 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.)

