1990 AMC 12 Problem 9

Attempt Problem 9 of the 1990 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1990 AMC 12 solutions, or check the answer key.

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Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is

2

3

4

5

6

Answer: B

Concepts:cube covering argument construction

Difficulty rating: 1450

Small hint:

Each edge belongs to exactly two faces

Big hint:

For the matching upper bound, choose three mutually nonadjacent edges

Solution:

Each black edge can cover only its two incident faces, so covering all

6

faces requires at least

3

black edges. Choose edges incident to the face-pairs top/front, bottom/left, and back/right. These three edges cover all six faces, so the minimum is

3.

Thus the correct answer is B.