Problem 4 · 1998 AJHSME
Easy
Geometry & Measurement
careful-counting
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Answer: E — 5 triangles.
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Hint 1 of 2
Counting by eye gives a guess; counting by SIZE gives the answer. Sort triangles into 'single smallest pieces' first, then ones made of two pieces, then the whole outline.
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Hint 2 of 2
Work small-to-big in layers so you never miss or double-count: how many 1-piece triangles? how many 2-piece? then the full outline.
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Approach: count in size layers (smallest pieces, then combinations)
- Smallest pieces: the inner line cut the figure into 3 little triangles.
- Made of 2 pieces: the two right-hand little triangles share a side and join into 1 bigger triangle.
- Made of all the pieces: the whole outline is itself 1 triangle.
- Total: 3 + 1 + 1 = 5.
- Why this transfers: for any 'how many triangles' puzzle, count by size layer (1 piece, 2 pieces, 3 pieces, …) instead of randomly. The layered list is your guarantee against missing or repeating one.
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