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?

$125$

$250$

$405$

$640$

$810$

Video solution:

Click to load, then click again to play

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.

2020 AMC 10A Problem 19

Video solution:

Written solution:

Problem 19 in Other Years