2016 AIME I Problems/Problem 3

Problem 3

A is a -faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.

Solution

Think about each plane independently. There are 5 ways to go from the first plane to the second plane. There are 9 ways to go horizontally around the second plane regardless of where you start-up to 4 to the right, up to 4 to the left, or not at all. Then, there are 2 paths down to the third plane. There are nine paths you can take here as well. Finally, there is only one way down to the bottom vertrex. Therefore there are paths.