2025 HMMT November General 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).

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2025 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.