HMMT 十一月 2024 · 冲刺赛 · 第 12 题

HMMT November 2024 — Guts Round — Problem 12

[8] A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to P . 10 9 4 P 5 3 2 8 1 7 6 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2024, November 09, 2024 — GUTS ROUND Organization Team Team ID#

解析

[8] A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to P . 10 9 4 P 5 3 2 8 1 7 6 11 Proposed by: Arul Kolla Answer: 5 Solution: The face opposite to P must be three faces away from P . It’s immediately clear that P is adjacent to 1 and 3. Also, P , 2 and 10 share a vertex. Furthermore, the faces around 10 are P , 2, 7, 8 and 9 in that order, so P is adjacent to 9. We conclude the neighbors of P are faces 3, 1, 2, 10, and