2024 AMC 8 Problem 17
Attempt Problem 17 of the 2024 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
17.
A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a grid attacks all other squares, as shown below. Suppose a white king and a black king are placed on different squares of a grid so that they do not attack each other. In how many ways can this be done?
Answer: E
Small hint:
Count ordered placements for the white king first
Big hint:
A corner allows safe black-king squares; an edge-center allows
Solution:
Count ordered placements by first choosing the square for the white king. If the white king is in the center, it attacks every other square, so there are choices for the black king.
If the white king is in a corner, it attacks squares, so the black king has safe squares. There are corner choices, giving placements.
If the white king is on an edge but not a corner, it attacks squares, so the black king has safe squares. There are such edge choices, giving placements.
The total number of placements is .
Thus, E is the correct answer.
Problem 17 in other years
- 1985 AMC 8
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