**Reference**:- Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman, “Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard”, in
*Proceedings of the the 31st International Symposium on Algorithms and Computation (ISAAC 2020)*, Hong Kong, December 14–18, 2020, 33:1–33:14. **Abstract**:-
We prove PSPACE-completeness of two classic types of Chess problems when
generalized to
*n*×*n*boards. A “retrograde” problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is “valid” or “legal” or “reachable”. Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A “helpmate” problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle. **Comments**:- This paper is also available from LIPIcs, and from arXiv.
**Updates**:- The figures of this paper use piece shapes from Wikimedia authored by Colin M.L. Burnett, licensed under BSD License. See SVG Tiler chess definitions for more details.
**Length**:- The paper is 14 pages.
**Availability**:- The paper is available in PDF (778k).
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Last updated August 19, 2022 by Erik Demaine.