Question

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

$20$

$24$

$27$

$28$

$32$

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 $0$ choices for the black king.

If the white king is in a corner, it attacks $3$ squares, so the black king has $9-1-3=5$ safe squares. There are $4$ corner choices, giving $4\cdot5=20$ placements.

If the white king is on an edge but not a corner, it attacks $5$ squares, so the black king has $9-1-5=3$ safe squares. There are $4$ such edge choices, giving $4\cdot3=12$ placements.

The total number of placements is $20+12=32$ .

Thus, E is the correct answer.

2024 AMC 8 Problem 17

Solution:

Problem 17 in Other Years