2024 AMC 8 Problems/Problem 17
Contents
[hide]Problem
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 x grid attacks all other squares, as shown below. Suppose a white king and a black king are placed on different squares of a x grid so that they do not attack each other. In how many ways can this be done?
(A) (B) (C) (D) (E)
Solution 1
Corners have spots to go and corners so . Sides have spots to go and sides so in total. is the answer.
~andliu766
Solution 2
We see that the center is not a viable spot for either of the kings to be in, as it would attack all nearby squares.
This gives three combinations:
Corner-corner: There are 4 corners, and none of them are touching orthogonally or diagonally, so it's
Corner-edge: For each corner, there are two edges that don't border it,
Edge-edge: The only possible combinations of this that work are top-bottom and left-right edges, so for this type
Multiply by two to account for arrangements of colors to get ~ c_double_sharp
Video Solution 1 (super clear!) by Power Solve
Video Solution 2 by Math-X (First understand the problem!!!)
~Math-X