The answer is 100 rectangles.
This one fools almost everybody, because most people start counting rectangles one at a time — and lose track long before they finish. Here’s the clean way to do it, the way that can’t miss any.
A rectangle in a grid is completely fixed by two choices: pick 2 of the vertical grid lines to be its left and right sides, and pick 2 of the horizontal grid lines to be its top and bottom. A 4×4 grid of unit squares has 5 vertical lines and 5 horizontal lines.
How many ways are there to choose 2 lines out of 5? Count them: 4 + 3 + 2 + 1 = 10 ways. Same for the horizontal lines: 10 ways. Every pairing of one vertical choice with one horizontal choice makes exactly one rectangle, and every rectangle comes from exactly one pairing. So: 10 × 10 = 100 rectangles. No counting them off one by one, no missing the sneaky wide ones hiding across the middle.
And here’s the bonus fact that starts arguments: 30 of those 100 are squares. Count them by size — 16 little 1×1 squares, 9 squares of 2×2, 4 squares of 3×3, and 1 big 4×4 square that is the whole grid. 16 + 9 + 4 + 1 = 30. Squares are rectangles, so they were inside your 100 all along — which means the other 70 are rectangles that aren’t squares.
Did you land on 100, or did you stop somewhere in the 40s and 50s like most people? Be honest in the comments — and watch our page for tomorrow’s counting game.
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