2024 AMC 8 Problems/Problem 17

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 $3$ x $3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3$ x $3$ grid so that they do not attack each other. In how many ways can this be done?

(A) $20$ (B) $24$ (C) $27$ (D) $28$ (E) $32$

Solution 1

Corners have $5$ spots to go and $4$ corners so $5 \times 4=20$ . Sides have $3$ spots to go and $4$ sides so $3 \times 4=12$ $20+12=32$ in total. $\boxed{\textbf{(E)} 32)}$ is the answer.