Complete Solutions for a Combinatorial Puzzle in Linear Time

TL;DR

This paper presents an optimal linear-time algorithm for solving a combinatorial puzzle involving shifting black and white checkers, providing explicit solutions and minimal step counts for general cases.

Contribution

The paper introduces a new optimal algorithm and explicit solutions for the checker shifting puzzle, achieving minimal steps in linear time.

Findings

Minimum steps for the game are nm + n + m.

An optimal move sequence can be generated efficiently.

Explicit solutions are provided for the general case.

Abstract

In this paper we study a single player game consisting of $n$ black checkers and $m$ white checkers, called shifting the checkers. We have proved that the minimum number of steps needed to play the game for general $n$ and $m$ is $nm + n + m$. We have also presented an optimal algorithm to generate an optimal move sequence of the game consisting of $n$ black checkers and $m$ white checkers, and finally, we present an explicit solution for the general game.