Problem 19 · 2011 AMC 8
Medium
Counting & Probability
systematic-counting
How many rectangles are in this figure?
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Answer: D — 11 rectangles.
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Hint 1 of 2
Eyeballing always under- or over-counts. Give every little region its own letter, then count by building rectangles out of those pieces — that way each one is found exactly once.
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Hint 2 of 2
Sweep by size: first the single-piece rectangles, then two-piece ones, then larger blocks. A group of pieces counts only if its outline is a true rectangle (four right-angle corners, no notches).
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Approach: label the pieces, then sweep size-by-size so nothing is missed or double-counted
- Cut the figure into its smallest regions and letter them. Now every rectangle in the picture is some block of these pieces, and you can list them in order of how many pieces they use.
- Single pieces that are themselves rectangles, then pairs that combine into a rectangle, then larger blocks — checking each candidate has a clean rectangular outline (no jagged edge).
- This organized sweep turns up exactly 11 rectangles in all.
- Worth keeping: ‘how many rectangles’ problems are won by a fixed order of counting (by size, by position), never by hunting at random — an order is what guarantees you hit each one once.
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