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This is a follow up to A knight chased by three knights

Two players are playing a variant of chess on a 11x11 grid. The first player controls a white knight that starts in the centre square. The second player controls four black knights that start in the corners. The first player aims to avoid being captured for as long as possible, while the second player aims to capture the white knight. Players alternate in making moves: the first player moves his white knight, then the second player moves all his black knights and so on. Passes are not allowed. The white knight is allowed to capture black knights.

If both players play optimally what would be the outcome of this game? Can white escape indefinitely? Can black guarantee to capture white no matter what he does?

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  • 3
    $\begingroup$ Congrats 10k rep! $\endgroup$ – Culver Kwan May 30 at 3:13
  • $\begingroup$ Thank you my friend! $\endgroup$ – Dmitry Kamenetsky May 30 at 4:38
  • $\begingroup$ Daniel I am really sorry, but I changed it to 11x11, because i thought 9x9 is too easy. Could you please bring back your 9x9 answer as I think it is still very interesting. You can keep it separate to the 11x11 answer. $\endgroup$ – Dmitry Kamenetsky May 30 at 4:40
  • $\begingroup$ To clarify, does Black get to move all of his knights on his turn, rather than having to choose one like in normal chess? If so, is it a requirement that they all move, and if two of them are a knight's move apart from each other, can they "swap" (effectively losing their move) if they're all moving at the same time? $\endgroup$ – Joseph Sible-Reinstate Monica May 30 at 22:15
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    $\begingroup$ Joseph good questions. I think in the interest of fairness, black has to move all his knights (they cannot pass). He cannot swap knights. $\endgroup$ – Dmitry Kamenetsky May 30 at 23:54
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This should be more or less

an easy win for the four black knights.

First of all,

white cannot ever capture. All the knights are on same coloured squares whenever it's white's turn.

Secondly, if black can ever reach this position

enter image description here

with white to move, then it's ggs on any finite sized board: black is guarding the marked squares, so white can only make a move toward the edge of the board, which both black knights will copy, and the position will repeat. Eventually white will run out of room to run, and will have to step into a threatened square and get captured.

There are several other positions that result in a similar endgame, for example:

enter image description here

this works too:

enter image description here

and so does this:

enter image description here

(These are not all of the "two-knight walls" that force a win, but this answer is getting needlessly repetitive already.)

Figuring out whether black can force white into one of these positions using only two knights is left as an exercise for the reader. With four knights, it's not much of an exercise.

| improve this answer | |
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  • $\begingroup$ Well done! This is a great answer. $\endgroup$ – Dmitry Kamenetsky Jun 1 at 0:44
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Note: The question was originally set on a 9x9 board. I posted an answer before noticing the change to 11x11. Presented here is an improved solution for the 9x9 board. While this does not directly answer the question in its current form, it should provide insight into larger boards and confirm the result that Bass gives:

With four knights, black can easily corner and capture the white knight in a finite number of moves.

In the following diagrams, W is the white knight, B is a black knight, o is a safe move for white, and x is an immediate loss for white.

Black can capture the white knight on turn three. There is no way for white to avoid this. (Except by being captured on turn two)

Position after move 1:

 . . o . x . . . .
 . o B . . o B . .
 . . . W . . . . .
 . x . . . x . . .
 . . x . o . . . .
 . . . . . . . . .
 . B . . . . . B .
 . . . . . . . . .
 . . . . . . . . . 

After move 2a:

 . . W . . . . . .
 x . . . x . . . .
 . x . x . . . . .
 . B . . . B . . .
 . . B . . . B . .
 . . . . . . . . .
 . . . . . . . . .
 . . . . . . . . .
 . . . . . . . . . 

After move 2b:

 . . . x . . . . .
 . W . . . . . . .
 . . . x B . . . .
 x . x B . . . . .
 . . B . . . . . .
 . . . . . B . . .
 . . . . . . . . .
 . . . . . . . . .
 . . . . . . . . . 

After move 2c:

 . . . x . . . x .
 . . . . . W . . .
 . . . x B . . x B
 . . . . x . x . .
 . . B . . . B . .
 . . . . . . . . .
 . . . . . . . . .
 . . . . . . . . .
 . . . . . . . . . 

After move 2d:

 . . . . . . . . .
 . . . . . . . . .
 . . . x . x . . .
 . . x B . B x . .
 . . . . W . . . .
 . . x B . B x . .
 . . . x . x . . .
 . . . . . . . . .
 . . . . . . . . . 

| improve this answer | |
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