2024 AMC 8 Problems/Problem 17
Contents
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 
Multiply by two to account for arrangements of colors to get 
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 by Math-X (First understand the problem!!!)
~Math-X
