Problem 23 · 2022 AMC 8
Hard
Counting & Probability
careful-countingcaseworksymmetry
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Answer: D — 84 configurations.
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Hint 1 of 2
Key observation: a ▵ line and a ○ line can't cross — the crossing cell would have to be both shapes. So the two lines can't be perpendicular: either both run across (rows) or both run down (columns). That splits the whole problem into two mirror-image halves.
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Hint 2 of 2
By that left-right/up-down symmetry, count only the “both vertical” configurations and double. Then case on how many full columns are monochromatic: all 3, or exactly 2.
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Approach: perpendicular lines are impossible ⇒ count one orientation and double, then casework
- Insight: where would a ▵-row and a ○-column meet? That cell can't be both shapes, so a horizontal line and a vertical line can't coexist. Hence both lines are horizontal OR both vertical — two symmetric worlds. Count the vertical world and multiply by 2.
- Vertical: case 3 lines — each of the 3 columns is monochromatic. 2³ = 8 colorings, minus the 2 all-same colorings (only one shape gets a line) → 6.
- Vertical: case 2 lines — one ▵ column, one ○ column, one mixed. 3 ways to pick the ▵ column × 2 remaining for ○ × 6 mixed configurations for the leftover column (2³ minus the two all-same = 6) = 36.
- Vertical total: 6 + 36 = 42. By symmetry, horizontal also = 42. Total: 42 + 42 = 84.
- You'll see this again: an impossibility (the lines can't be perpendicular) is what splits the count into clean, non-overlapping cases — and a symmetry then lets you count one case and double. Always check that the cases can't overlap (a config with both a vertical and horizontal monochrome line would be double-counted — here that's exactly what the impossibility rules out).
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