Math Gold Medalist

Lor

2024 AMC 8 

Problem 17

A chess king is said to attack all 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 3 x 3 grid so that they do not attack each other. In how many ways can this be done?

[Insert Diagram]

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32$

Casework

Consider cases of white king

If white king is on the corner then black king has 5 cases

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   Solution