Chess is hard even for a single player

@inproceedings{Aravind2022ChessIH,
  title={Chess is hard even for a single player},
  author={N. R. Aravind and Neeldhara Misra and Harshil Mittal},
  booktitle={FUN},
  year={2022}
}
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 × 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… 

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For the other direction, note that by Claim 18, we have that (T , v) is 0-solvable and in particular
    Let v be the vertex where the final capture is made in a valid capture sequence. Then (T , v) is 0-solvable, and by Claim 18 every internal node of (T , v) has a leaf neighbor
      Suppose that (T , v) is 1-solvable. Then (T , v) is 0-solvable and further, either |T | = 1, or by Claim 18
        Suppose that (T , v) is 1-solvable. Then (T , v) is 0-solvable and further, either |T | = 1, or by Claim 17
          T , v) is 1-solvable ⇐⇒ |T | = 1 OR (T , v) is 0-solvable and v has a leaf neighbor. Proof. We prove (a) and (b) together by induction on
            = 1 and |V(T ) = 2, the statements are easily verified to be true
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