The answer is 150 rectangles.
This grid looks harmless — 4 columns by 5 rows of little rectangles, just 20 cells. But a rectangle doesn’t have to be one little cell. Any rectangle whose edges sit on the grid lines counts, of EVERY width and EVERY height. And this grid is drawn with 5 vertical lines and 6 horizontal lines — which gives us a proof, not a guess.
Here’s the clean way to see it: every rectangle is completely decided by picking 2 of the 5 vertical lines (its left and right edges) and 2 of the 6 horizontal lines (its top and bottom edges). Choose 2 from 5 and you get 10 possible widths-and-positions. Choose 2 from 6 and you get 15 possible heights-and-positions. Multiply: 10 × 15 = 150.
And here’s the size-by-size breakdown, so you can check every single one:
- 1 cell wide: 20 (1×1) + 16 (1×2) + 12 (1×3) + 8 (1×4) + 4 (1×5) = 60
- 2 cells wide: 15 (2×1) + 12 (2×2) + 9 (2×3) + 6 (2×4) + 3 (2×5) = 45
- 3 cells wide: 10 (3×1) + 8 (3×2) + 6 (3×3) + 4 (3×4) + 2 (3×5) = 30
- 4 cells wide: 5 (4×1) + 4 (4×2) + 3 (4×3) + 2 (4×4) + 1 (4×5) = 15
Add them up: 60 + 45 + 30 + 15 = 150. Notice the pattern that proves it as you go: each time a rectangle gets one cell wider, it loses one possible starting column; each time it gets one cell taller, it loses one possible starting row. The two traps are stopping at the 20 single cells — and forgetting the biggest rectangle of all, the whole grid itself, which is hiding in that breakdown as the single 4×5.
Did you land on 150, or did the big ones slip past you? Be honest in the comments — and watch our page for tomorrow’s counting game.
