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Fog of War Chess

Matthias Gehnen, Julius Stannat

TL;DR

Fog-of-War chess imposes imperfect information and restricts visibility, motivating a theoretical endgame analysis. The paper analyzes three endgames: $KQ$ vs $K$, $KR$ vs $K$, and $KRR$ vs $K$, establishing that $KQ$ always wins, $KR$ does not always guarantee a win, and $KRR$ always wins, with constructive strategies and lemmas guiding the proof. The $KQ$ endgame uses a two-stage approach—achieving a corner configuration and then pushing the Black king to the edge—while the $KR$ endgame reveals inherent limitations, and the $KRR$ endgame demonstrates a staircase-like mating method. These results extend endgame theory under imperfect information and suggest directions for AI planning in Fog-of-War settings, though many open questions remain for other piece combinations and optimization under 50-move-like constraints.

Abstract

Fog of War chess is a popular variant of classical chess, in which both players have only partial information about the position of the opponent's pieces. This study provides the first theoretical analysis of endgames in Fog of War chess. In particular, we analyze the setups king and queen versus king, king and rook versus king, and king and two rooks versus king. We show that a king and queen can always guarantee a win against a lone king. In contrast to classical chess, a king and a rook cannot guarantee a win against a lone king. However, adding one more rook guarantees a win.

Fog of War Chess

TL;DR

Fog-of-War chess imposes imperfect information and restricts visibility, motivating a theoretical endgame analysis. The paper analyzes three endgames: vs , vs , and vs , establishing that always wins, does not always guarantee a win, and always wins, with constructive strategies and lemmas guiding the proof. The endgame uses a two-stage approach—achieving a corner configuration and then pushing the Black king to the edge—while the endgame reveals inherent limitations, and the endgame demonstrates a staircase-like mating method. These results extend endgame theory under imperfect information and suggest directions for AI planning in Fog-of-War settings, though many open questions remain for other piece combinations and optimization under 50-move-like constraints.

Abstract

Fog of War chess is a popular variant of classical chess, in which both players have only partial information about the position of the opponent's pieces. This study provides the first theoretical analysis of endgames in Fog of War chess. In particular, we analyze the setups king and queen versus king, king and rook versus king, and king and two rooks versus king. We show that a king and queen can always guarantee a win against a lone king. In contrast to classical chess, a king and a rook cannot guarantee a win against a lone king. However, adding one more rook guarantees a win.
Paper Structure (10 sections, 22 theorems, 2 figures)

This paper contains 10 sections, 22 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Rules
  3. Notation
  4. Queen and King vs. King
  5. Preparing Observations
  6. Stage 1: Achieving the corner configuration
  7. Stage 2: Push the Black king to the edge, line by line
  8. Rook and King vs. King
  9. Two Rooks and King vs. King
  10. Final Comments and Open Questions

Key Result

Lemma 1

If the queen is adjacent to her own king, the queen can reach any other square adjacent to the king. If the queen gets captured, White wins immediately.

Figures (2)

  • Figure 1: White queen on A---[] 1---[] A[] 1[] 1 1 $A1$$(\Xsub,\Ysub)$$(\Xsub,\Ysub)$ , Black king on H---[] 8---[] H[] 8[] 1 1 $H8$$(\Xsub,\Ysub)$$(\Xsub,\Ysub)$ , Black rook on E---[] 4---[] E[] 4[] 1 1 $E4$$(\Xsub,\Ysub)$$(\Xsub,\Ysub)$
  • Figure 9: The green squares mark the safe space C-F---[] 3-6---[] C-F[] 3-6[] 1 1 $(\Xsub,\Ysub)$$(\Xsub,\Ysub)$$(\Xsub,\Ysub)$ . The orange squares indicate the danger-zone, the red squares indicate the death-zone

Theorems & Definitions (22)

  • Lemma 1
  • Lemma 2
  • Lemma 3: Ensuring a Safe Way
  • Lemma 4: Move away from the center
  • Lemma 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 12 more