A study of a generalization of a card problem

Abstract

In this paper, we study a card problem, and its generalization, related to a computer based Cribbage player, namely, how many ways are there to select 12 Poker cards out of a deck of 52 such that their total face value exceeds 31? As an immediate application, the answer to the above question provides useful information for designing and enhancing computer based game playing software for Cribbage and other similar card games. Moreover, the analysis of the general problem also leads to a solution to a problem of “sampling without replacement”. Besides presenting a combinatorial solution to the general problem, and discussing its “expressive complexity” via its relationship with the well-known integer partition problem, we also provide a computer solution, implemented in Prolog, based on a non-trivial recurrence relation. Relying on the latter result, we then carry out preliminary analysis of the computational aspects of some of the rules governing the Cribbage game.