How to solve figure counting puzzles
Count by size class, and nothing escapes
'How many squares?' means all sizes, including the composites. Enumerate one size class at a time and the count becomes arithmetic, not squinting.
Updated
'How many squares in this figure?' The planted error is counting only the unit cells and missing the composites — the 2×2s, the 3×3s, the whole outer square. Trained counting never sweeps the figure hoping to notice everything; it walks size classes in order, counting each exhaustively.
Squares in a grid: sum of squares
An n×n grid holds n² unit squares, (n−1)² of size 2, down to 1 of size n: a 3×3 grid has 9+4+1 = 14. A p×q rectangle-count uses line-pairs: choose 2 of the p+1 vertical and 2 of the q+1 horizontal lines — every choice is a rectangle. Knowing the two formulas turns the hardest counting items into ten-second arithmetic.
Triangles: class by size and orientation
In triangular grids, count upward-pointing triangles by size, then downward-pointing by size — the downward ones are the universally forgotten class. Composite triangles built from four or nine small ones exist in both orientations.
Marked-cell counts: contains vs avoids
'How many rectangles contain the shaded cell?' — multiply choices on each side of it (lines left × lines right × lines above × lines below). 'How many avoid it?' — total minus containing. Convert the question to arithmetic before counting anything by eye.
The strategy
- Say the size classes aloud before counting ('1×1, 2×2, 3×3, done').
- Count each class in reading order — a fixed scan path prevents doubles.
- Add the classes; sanity-check against the formula when the figure is a clean grid.
- Odd figures with holes: count the clean grid, then subtract what the hole destroys.
Common traps
- Unit-only counts — the single most farmed distractor in this family.
- Double counting from an undisciplined scan path.
- Downward triangles and tilted squares (corners on midpoints) — the invisible classes.
- Formula applied to a figure that isn't actually a full grid — check for missing cells first.
Speed · ~45s/item: Class-by-class is 30–60 seconds even on 4×4 figures, and the formulas compress clean grids to ten.
Try it yourself: 4 practice questions
Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.
How many rectangles does the figure contain, counting every size?
Reveal the answer
The answer is C — 60.
Show why — the worked reasoning
How many squares does the figure contain, counting every size?
Reveal the answer
The answer is A — 14.
Show why — the worked reasoning
How many squares does the figure contain, counting every size?
Reveal the answer
The answer is B — 8.
Show why — the worked reasoning
How many squares does the figure contain, counting every size?
Reveal the answer
The answer is D — 26.
Show why — the worked reasoning
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Questions people ask
Will figure counting questions be on the free test?
Maybe. The free test gives spatial reasoning 4 of its 20 questions, spread across 9 different question types, so any one test may or may not include this kind.