Question

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

$3$

$5$

$4$

$8$

$6$

Answer: C

Concepts:combinatorial game extremal argument

Difficulty rating: 1560

Solution:

Suppose every guess misses. Then the domino lies entirely on unguessed squares, so those squares include two adjacent cells. To force a hit, we need the unguessed squares to have no two adjacent. Color the grid like a checkerboard. The biggest set of pairwise non-adjacent squares has $5$ cells, one color's worth: the four corners and the center. So we must guess at least $9 - 5 = 4$ squares. And $4$ is enough: guess the four edge midpoints, since every domino covers one square of each color, hence one edge midpoint. Therefore, the answer is C.

2023 AMC 10B Problem 10

Answer: C

Difficulty rating: 1560

Solution:

Problem 10 in Other Years