MinDist is less than 7
Purushottam Saha, Diganta Mukherjee
TL;DR
MinDist measures the minimum number of card replacements needed to transform a 13-card Rummy hand into a declarable meld partition in the presence of a wildcard joker. We sharpen the universal upper bound from $9$ to $7$ by a synthesis of elementary combinatorial arguments on suit distributions, gaps within suits, and forced meld structures, supplemented by targeted computational verification for residual cases. A key lemma shows that if a suit contains at least $4$ cards, the maximum possible minimum gap between two cards is $2$, which constrains meld construction and drives the $7$-card replacement bound. We also exhibit an explicit extremal hand that attains MinDist $7$ with a suitable wild-card joker, and verify the remaining configurations via computer search (code available at the linked repository). These results provide a complete extremal characterization of MinDist and have practical implications for algorithmic hand evaluation and worst-case analysis in automated Rummy play; directions for future work include rule variations and multiple jokers.
Abstract
The metric MinDist, introduced recently to quantify the distance of an arbitrary Rummy hand from a valid declaration, plays a central role in algorithmic hand evaluation and optimal play. Existing results show that the MinDist of any $13$-card Rummy hand from a single deck is bounded above by $9$. In this paper, we sharpen this bound and prove that the MinDist of any hand is at most $7$. We further show that this bound is tight by explicitly exhibiting a hand whose MinDist equals $7$ for a suitable choice of wildcard joker. The proof combines elementary combinatorial arguments with structural properties of card partitions across suits and resolves the gap between the previously known upper bound and the true extremal value.