Ice sliding games

TL;DR

This paper investigates sliding games where a robot moves until hitting an obstacle, determining the minimum number of blocks needed for full coverage on grids and tori, and explores various game variants.

Contribution

It provides exact minimum block counts for complete coverage on rectangular grids and tori, and analyzes different game variants.

Findings

Exact minimum blocks for grid coverage

Exact minimum blocks for torus coverage

Analysis of game variants with walls and stopping constraints

Abstract

This paper deals with sliding games, which are a variant of the better known pushpush game. On a given structure (grid, torus...), a robot can move in a specific set of directions, and stops when it hits a block or boundary of the structure. The objective is to place the minimum number of blocks such that the robot can visit all the possible positions of the structure. In particular, we give the exact value of this number when playing on a rectangular grid and a torus. Other variants of this game are also considered, by constraining the robot to stop on each case, or by replacing blocks by walls.