PUMaC 2025 · 组合（B 组） · 第 1 题

PUMaC 2025 — Combinatorics (Division B) — Problem 1

There is an alphabet consisting of four shapes; a Segment, a Diamond, a Triangle, and a Pentagon. All of these shapes are regular. Each shape has a marked vertex that indicates its current orientation (as shown below). Richard creates a password of length 6 from this four-shape alphabet. Then, every second, ◦ ◦ ◦ Richard rotates each Segment by 180 , each Diamond by 90 clockwise, each Triangle by 120 ◦ clockwise, and each Pentagon by 72 clockwise. It turns out that the shapes are first all oriented the same as in his original password after 20 seconds. How many distinct passwords could Richard have chosen?

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