Chess is hard even for a single player

TL;DR

The paper studies a single-player chess variant called Solo Chess, proves its computational complexity as NP-complete in general, and characterizes solvable cases on specific boards, also introducing a related graph game.

Contribution

It generalizes Solo Chess to unbounded boards with multiple pieces, proves NP-completeness for certain configurations, and characterizes solvable instances and related graph capture games.

Findings

Generalized Solo Chess is NP-complete for rooks and queens with limited captures.

Solvable instances are characterized for 1D rook and 2D pawn cases.

Graph Capture Game is NP-complete for undirected graphs and DAGs.

Abstract

We introduce a generalization of "Solo Chess", a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 x 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e, make a sequence of captures after which only one piece is left. We generalize this game to unbounded boards with $n$ pieces, each of which have a given number of captures that they are permitted to make. We show that Generalized Solo Chess is NP-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be NP-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of Generalized Solo Chess can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board. Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game, which involves clearing a graph of tokens via captures along edges. This game subsumes Generalized Solo Chess played by knights. We show that the Graph Capture Game is NP-complete for undirected graphs and DAGs.