Rectangles on a Chessboard

Question

How many rectangles (of any size, aligned with the squares) can be outlined on a standard 8×8 chessboard?

Accepted Answer

Rectangles are pairs of lines. The board's grid has 9 horizontal and 9 vertical lines; an axis-aligned rectangle is exactly a choice of 2 horizontal lines (top and bottom) and 2 vertical lines (sides): binom92^2 = 36^2 = 1296. The bijection does all the work, no size case-bash, no double counting. Approach 2, sum over dimensions A w times h rectangle can sit in (9-w)(9-h) positions, so the total is sum_w=1^8sum_h=1^8(9-w)(9-h) = left(sum_w=1^8(9-w)right)^2 = 36^2 = 1296, the double sum factorizes, which is Approach 1 rediscovered. Squares need w = h, breaking the factorization: sum_k=1^8(9-k)^2 = 64 + 49 + dots + 1 = 204. The trap: answering 64 (unit squares) or 204 (all squares) to the rectangle question, listen for which object is asked. The line-choice frame also answers "how many rectangles contain a given cell" and generalizes instantly to m times n boards: binomm+12binomn+12.

Rectangles on a Chessboard

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