How Many Rectangles Are There?

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Find a general formula involving and for the total number of rectangles. There is a second hint in the details, but do not look too quickly—thinking about the solution is more important than the answer. Note that a square is a special kind of rectangle, so count them too.

Contributed by: Abraham Gadalla (March 2011)
Open content licensed under CC BY-NC-SA



Hint 1: Click the hint checkbox.

Hint 2: Choose an horizontal rectangle on the bottom and a vertical rectangle on the left. Add all the rectangles above to form a green strip across the big rectangle . Similarly, add all the rectangles to the right of to form a red strip across . The two strips overlap to form a blue rectangle . and determine uniquely and vice versa. To find how many ways there are to form a blue rectangle, count the number of pairs .

Hint 3: In a row of length there are rectangles.

This problem generalizes to how many rectangular sub-boxes can be made from a three-dimensional rectangular box of dimensions , where , , and are positive integers. Sketch a hint and derive the formula.

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