Beyond Nine Gates: Classification of Full-Sided Mahjong Waits with $3n$ Ranks
Shubin Xue, Xingyu Zhong
math.GM
Nov 2, 2026 · v1
TL;DR
Results are partially supported by an ongoing Lean 4 formalization of the Mahjong full-sided wait classification, with a public GitHub repository.
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.
Problem
Folklore holds that Nine Gates is the unique single-suit 13-tile Mahjong hand that wins on any drawn rank. Earlier confirmations relied on exhaustive computer enumeration over fixed finite configurations.
Approach
The authors build a combinatorial framework for single-suit Mahjong with 3n ranks, a variable number of melds, and a variable multiplicity bound. It uses virtual hands, meld placements, congruence-class invariants and a draw-pair graph. Interior positivity and splitting at missing ranks then yield a classification proof that needs no enumeration. Part of the work is being formalized in Lean 4, with some candidate formalizations drafted with AI assistance.
Results
For every n≥2 there are exactly 8 full-sided waits with n+1 melds, and Nine Gates is the unique live (valid) one. The case m>n+1 is posed as an open problem, together with partial results on abundance and on minimal waits, including the maximum meld count of minimal full-sided waits.
