Competition problem — Combinatorics

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Combinatorics Difficulty 4.9 AIME Find the answer United States

Problem:

You have a die with faces labelled $1$ through $6$ . On each face, you draw an arrow to an adjacent face, such that if you start on a face and follow the arrows, after $6$ steps you will have passed through every face once and will be back on your starting face. How many ways are there to draw the arrows so that this is true?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Answer: $32$

There are $4$ choices for where to go from face $1$ . Consider the $4$ faces adjacent to $1$ . We can visit either $1$ , $2$ , or $3$ of them before visiting the face opposite $1$ . If we only visit one of these adjacent faces, we have $4$ choices for which one, then we visit face $6$ , opposite face $1$ , then we visit the remaining $3$ faces in one of two orders—for a total of $8$ ways. If we visit $2$ adjacent faces first, there are $8$ choices for these two faces, then $2$ choices for the path back from face $6$ to face $1$ . Lastly, there are $8$ ways to visit three of the adjacent faces before visiting the opposite face. These choices give $32$ total.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.