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Alice and Bob play a special chess game with alternate moves. Alics starts with an empty chessboard and puts a Queen at a field in the first row (A8-H8) or the right column (H1-H8). Now Bob is moving the Queen according to the chess rules, however with the following additional limitations:
1) A horizontal move is only allowed to the left
2) A vertical move is only allowed to the bottom
3) A diagonal move is only allowed to left bottom

Bob makes the first move. The winner is the person, who placed the Queen at the bottom left corner (A1).

Who has a winning strategy?

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  • $\begingroup$ Do you mean toward instead of to in the rules? As I read them a vertical move must go all the way down to the first rank and a diagonal move is not allowed unless it ends at a1. "to the left" can mean all the way to the left or in that direction. $\endgroup$
    – Ross Millikan
    Mar 11, 2020 at 0:08

2 Answers 2

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We can work backwards to figure this out:

The bottom left space is a winning space: if you place the queen there, you win.
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So if you place the queen somewhere where the opponent can get to a winning space, you lose.
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And if you place the queen somewhere where the opponent must move to a losing space, you win.
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And if you place the queen somewhere where the opponent can move to a winning space, you lose.
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...and if your opponent must move to a losing space, you win...
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...and if your opponent can move to a winning space, you lose...
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...and if your opponent must move to a losing space, you win.
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Conclusion:

Alice can win the game by choosing to start on e8 (or h5); after that, on each of her turns she can always move to a green square, and Bob never can. So Alice will be the one to move to the bottom left corner and win.

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    $\begingroup$ I'm pretty sure I've seen -- in fact pretty sure I've answered -- a puzzle more or less equivalent to this one some time in the last year or thereabouts, but I can't find it at all... $\endgroup$
    – Gareth McCaughan ♦
    Mar 9, 2020 at 21:55
  • $\begingroup$ @GarethMcCaughan Yeah, I strongly suspected it was a duplicate as well but couldn't find the original. $\endgroup$
    – Deusovi ♦
    Mar 9, 2020 at 21:57
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    $\begingroup$ Incidentally, for anyone curious about this game it's usually called rot13(Jlgubss'f tnzr) and the pattern of winning and losing positions has a nice mathematical property. $\endgroup$
    – Gareth McCaughan ♦
    Mar 9, 2020 at 21:58
  • $\begingroup$ very nice presentation of the solution.. and very fast! $\endgroup$
    – ThomasL
    Mar 9, 2020 at 22:13
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    $\begingroup$ @PaulMcCarthy, if I got this right on a quick drawing, with a King, the player placing the piece on the border, anywhere, loses. Also the pattern of losing and winning squares is much more boring, just a regular grid with two steps horizontally and vertically between the winning squares. $\endgroup$
    – ilkkachu
    Mar 11, 2020 at 9:10
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Who ever plays first:

Will win if they place the queen on e8 or h5 (followed by perfect play).

Because:

a1, b3, c2, d6 and f4 are all winning squares that can be reached (only by the first player) after any 2nd, 4th or 6th move.

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