2020 AMC 10A Problem 19

Question

As shown in the figure below, a regular dodecahedron (the polyhedron consisting of

12

congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?

125

250

405

640

810

Answer: E

Concepts:graph theory multiplication principle casework

Difficulty rating: 2460

Small hint:

Because upward moves are forbidden, split every path into a top-ring phase and a bottom-ring phase

Big hint:

On a five-cycle, a self-avoiding walk has

1+2\cdot4

possible stopping paths

Video solution:

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Written solution:

After leaving the top face, choose one of the

5

top-ring faces. Because moves from the bottom ring to the top ring are forbidden, every valid path has a top-ring phase, then one move down to the bottom ring, then a bottom-ring phase.

Fix the first top-ring face. On the top ring, the path can move around the

5

-cycle without revisiting a face and then stop at any point: there are

1+2\cdot4=9

possible top-ring paths. From the stopping face, there are

2

possible downward moves to the bottom ring, so the top part has

18

choices.

Once in the bottom ring, the path can move around the bottom

5

-cycle without revisiting a face and then enter the bottom face; this gives

1+2\cdot4=9

choices. The total is

5\cdot18\cdot9=810

. Thus, E is the correct answer.