2023 AMC 12B Problem 5

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: 1350

Small hint:

Think about which squares you could leave un-guessed without ever touching the domino

Big hint:

The un-guessed squares must contain no two adjacent squares; the largest such set in a

3\times 3

grid has

5

squares

Solution:

A set of guessed squares is guaranteed to hit the domino if and only if the un-guessed squares contain no two adjacent squares, since otherwise the domino could hide on that adjacent pair. The largest set of pairwise non-adjacent squares in the

3\times 3

grid is the

5

-square checkerboard (four corners plus the center). So at most

5

squares can be left unguessed, and you must guess

9-5=4.

Thus, the correct answer is C.