HMMT 十一月 2025 · GEN 赛 · 第 6 题

HMMT November 2025 — GEN Round — Problem 6

Kelvin the frog is at the point (0 , 0 , 0) and wishes to reach the point (3 , 3 , 3). In a single move, he can either increase any single coordinate by 1, or he can decrease his z -coordinate by 1. Given that he cannot visit any point twice, and that at all times his coordinates must all stay between 0 and 3 (inclusive), compute the number of distinct paths Kelvin can take to reach (3 , 3 , 3).

解析

Kelvin the frog is at the point (0 , 0 , 0) and wishes to reach the point (3 , 3 , 3). In a single move, he can either increase any single coordinate by 1, or he can decrease his z -coordinate by 1. Given that he cannot visit any point twice, and that at all times his coordinates must all stay between 0 and 3 (inclusive), compute the number of distinct paths Kelvin can take to reach (3 , 3 , 3). Proposed by: Derek Liu 6 6 Answer: 4 = 81920 3 Solution: Projecting the path down to the xy -plane, we get an up-right path from (0 , 0) to (3 , 3), of 6 which there are = 20. Fix such an up-right path. Each edge in this path can be the projection 3 6 of four different edges Kelvin can take (at z = 0, 1, 2, and 3), so there are 4 ways to pick the six edges that project down to the up-right path. These can be connected into a unique path with vertical 6 6 moves, so the answer is 4 = 81920 . 3 © 2025 HMMT