Beyond Nine Gates: Classification of Full-Sided Mahjong Waits with $3n$ Ranks

Abstract

Mahjong is a traditional tile-based game popular in East and Southeast Asia. Combinatorial study of Mahjong hands has been a topic of interest in recreational mathematics, with the Nine Gates hand being a well-known example. This single-suit Mahjong hand $\{1,1,1,2,3,4,5,6,7,8,9,9,9\}$ consists of $13$ tiles of a single suit ranked $1$ through $9$, arranged such that drawing any tile from that suit completes a winning hand of a pair and $4$ melds, regardless of the tile's rank. Folklore has circulated that Nine Gates is the unique single-suit hand with this $9$-sided-wait property. While prior works have verified this uniqueness by exhaustive computer enumeration across different finite configurations, we present a general classification of full-sided waits for Mahjong variants with $3n$ ranks and $n+1$ melds. For every $n\geq2$, we report exactly $8$ full-sided waits, of which Nine Gates is the unique live one. Notably, our proof requires no exhaustive enumeration or any prior knowledge of the classification result. To address the problem in its right generality, a comprehensive combinatorial framework is developed, allowing the number of ranks, the number of melds, and the upper bound on tile multiplicities to vary independently. Reliability of our results is partially supported by a manually curated ongoing formalization in Lean 4.