Problem 17 · 2024 AMC 8
Hard
Counting & Probability
caseworkcareful-counting
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Answer: E — 32 ways.
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Hint 1 of 2
First, kill the impossible square: the CENTER attacks all 8 others, so a king there always attacks — neither king can sit there. That leaves only the 8 border squares.
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Hint 2 of 2
Technique — casework by the first king's spot, because corners and edge-middles see different numbers of squares: a corner attacks 3 (5 safe for the other), an edge-middle attacks 5 (3 safe).
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Approach: casework on the first king's position
- Rule out the center first: it attacks all 8 surrounding squares, so a king there can never avoid attacking — both kings live on the 8 border squares. The kings are different colors, so order matters; place white first, then count safe spots for black.
- White on a corner (4 corners): a corner attacks only 3 squares, leaving 8 − 3 = 5 safe for black. 4 × 5 = 20.
- White on an edge-middle (4 of them): it attacks 5 squares, leaving 3 safe. 4 × 3 = 12.
- Total: 20 + 12 = 32. Why split into cases: the count of attacked squares depends on the piece's position, so group positions by that count — the same move pays off whenever a board has corner/edge/center symmetry.
Another way — all pairs minus attacking pairs:
- Place white anywhere (9), then black on any other square (8): 9 × 8 = 72 ordered placements ignoring attacks.
- Subtract attacking placements. Count adjacencies on the 3×3: there are 12 edge-adjacencies and 8 diagonal-adjacencies, 20 unordered attacking pairs, so 40 ordered ones.
- Non-attacking = 72 − 40 = 32.
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