Problem 2

A prism with pentagons

A_1A_2A_3A_4A_5

and

B_1B_2B_3B_4B_5

as top and bottom faces is given. Each side of the two pentagons and each segment

A_iB_j

is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of different colors. Prove that all 10 sides of the top and bottom faces have the same color.

Step 1 of 4: Force the upper pentagon to be monochromatic

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A_1A_2\text{ and }A_1A_5\text{ have opposite colors}

Detailed analysis

Assume two edges incident with A1 have opposite colors. Among the five edges A1Bi, at least three share a color; one of the two incident upper edges has that same color. Use that edge and the three chosen cross-edges.