MinDist is less than 7

TL;DR

This paper proves that the maximum MinDist for any Rummy hand is 7, improving previous bounds and demonstrating the tightness of this limit through explicit examples.

Contribution

It sharpens the upper bound of MinDist for Rummy hands from 9 to 7 and confirms this bound is tight with explicit constructions.

Findings

Max MinDist for Rummy hands is 7

Constructed example with MinDist exactly 7

Improved theoretical understanding of Rummy hand distances

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.