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
Snapshots
Details
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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