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A bishop and a rook, of opposing colors, start on opposite corners on the same side of a chessboard, oriented as in the following picture:

Chess Board

They then proceed to play a game of cat-and-mouse.

Rules

  • The pieces make a move based on the rules of chess, meaning that the bishop travels along the diagonal while the rook travels along the parallels.
  • When a square is moved across, it becomes uninhabitable and can no longer be moved across.
  • The bishop always goes first.
  • If a piece is either a) captured, or b) does not move on its turn, whether by choice or because there are no valid moves left, that piece loses.

Based on these rules and assuming optimal play, which piece is crowned the victor?


Bonus Question 1: The same rules apply, except now a tile only becomes uninhabitable when a piece finishes its move on that tile. Still assuming optimal play, which piece wins this variation?

Bonus Question 2: If the bishop starts in the top lefthand corner instead of the bottom lefthand corner, and the rook is now allowed to go first, which piece is crowned the victor? (Answer this question for both variations of uninhabitable squares.)


Note: The open-ended tag is included because there is ambiguity associated with the restriction of "optimal play". A perfect answer will not only detail the victory condition of the winning piece, but also detail why the plays made can be considered "optimal" for each piece.

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  • 3
    $\begingroup$ Which is the cat? $\endgroup$ – warspyking Aug 24 '15 at 17:59
  • $\begingroup$ @warspyking er...both? I guess it's more accurately "cat-and-cat" or something, but that's not as catchy. $\endgroup$ – Bailey M Aug 24 '15 at 18:00
  • 2
    $\begingroup$ "Doctor Venture and I have been engaged in a deadly game of cat and also-cat for years!" -- The Monarch $\endgroup$ – dennisdeems Aug 24 '15 at 19:08
  • 2
    $\begingroup$ Why did you put the bishop in black if it always goes first? $\endgroup$ – Pierre Arlaud Aug 24 '15 at 21:26
  • $\begingroup$ @PierreArlaud cuz I'm stupid lol $\endgroup$ – Bailey M Aug 25 '15 at 0:17
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General thoughts

The rook has obviously an advantage here, because it can use black as well as white squares, while the bishop can only use black squares. Apart from that several squares are dead ends for the bishop like for example h8. If the rook can fence in the bishop in about a half of the board ("Tron-Principle™") it is almost always a guaranteed win.

Base Question

Complete winning strategy (in all cases the rook has more moves left than the number specified for the bishop):

1. ... Bg7 2. Rh5 (Bishop has 11 moves left)

1. ... Bf6 2. Rh7
    2. ... Bg5 3. Rh8 (Bishop has 6 moves left)
    2. ... Bd8 3. Rg7 (Bishop has 8 moves left)

1. ... Be5 2. Rh6
    2. ... Bf4 3. Re6 (Bishop has 5 moves left)
    2. ... Bg3 3. Re6 (Bishop has 4 moves left)
    2. ... Bg7 3. Rg6 (Bishop has 11 moves left)
    2. ... Bc7 3. Rh8 (Bishop has 4 moves left)
    2. ... Bb8 3. Rh7 (Bishop has 5 moves left)

1. ... Bd4 2. Rh5
    2. ... Be3 3. Rd5 (Bishop has 5 moves left)
    2. ... Bf2 3. Rd5 (Bishop has 2 moves left)
    2. ... Bg1 3. Rd5 (Bishop has 0 moves left)
    2. ... Bf6 3. Rh7 (2nd case above)
    2. ... Bg7 3. Rg5 (Bishop has 11 moves left)
    2. ... Bb6 3. Rh7
        3. ... Bd8 4. Rd7 (Bishop has 0 moves left)
        3. ... Ba5 4. Rh8 (Bishop has 2 moves left)
    2. ... Ba7 3. Rh8 (Bishop has 0 moves left)

1. ... Bc3 2. Rh5
    2. ... Bd2 3. Ra5 (Bishop has 6 moves left)
    2. ... Bf1 3. Ra5 (Bishop has 5 moves left)
    2. ... Bd4 3. Ra5 (Bishop has 6 moves left)
    2. ... Bf6 3. Rh7 (2nd case above)
    2. ... Bg7 3. Rg5 (Bishop has 11 moves left)
    2. ... Bb4 3. Rb5 (Bishop has 1 move left)

1. ... Bb2 2. Rh3
    2. ... Bc1 3. Rb3 (Bishop has 4 moves left)
    2. ... Bd4 3. Rh5 (4th case above)
    2. ... Be5 3. Rh6 (3rd case above)
    2. ... Bf6 3. Rh7 (2nd case above)
    2. ... Bg7 3. Rh5 (1st case above)

Bonus Question 1

I don't have a complete winning strategy here. But it seems even more obvious that the rook has an advantage here as well. My general strategy would be:
1. Block the black centre squares.
2. Block the path of the bishop on the sides (if the black centre squares are blocked, ideally only 1 more black square is needed for that).
3. Create dead ends for the bishop if possible.

Bonus Question 2

Complete winning strategy:

1. Re1 (as suggested by JonTheMon)

    1. ... Bb7/Bc6/Bd5 2. Re8 (Bishop has 13/9/9 moves left)

    1. ... Bf3 2. Re3 (Bishop has 12 moves left)

    1. ... Bg2 2. Re2 (Bishop has 11 moves left)
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  • 1
    $\begingroup$ So to be clear about what I'm looking at, the rook basically just stall out the bishop until it gets stuck in each of these scenarios, correct? $\endgroup$ – Bailey M Aug 24 '15 at 20:29
  • $\begingroup$ Yes, to be more specific, divide the board in 2 areas using the uninhabitable squares so that both pieces can't affect each other anymore. In each case the rook has more room to breathe. $\endgroup$ – Sleafar Aug 24 '15 at 20:35
  • $\begingroup$ Why would the bishop move to G7 on the first move? Surely the best strategy for the bishop is to leave as many moves available as possible by moving to B2. $\endgroup$ – Gordon K Aug 24 '15 at 21:10
  • $\begingroup$ @GordonK That is only one of the 6 possible moves of the bishop in the first turn. The other possible moves are listed in the variants below. Of course I ignored Bh8. $\endgroup$ – Sleafar Aug 24 '15 at 21:13
  • $\begingroup$ @Sleafar it isn't clear to me what you mean when you use the word "field". I haven't come across that term in my chess reading. $\endgroup$ – dennisdeems Sep 1 '15 at 13:31
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Base Question

Answer:

the Rook wins

Strategy:

In general a good winning strategy for the Rook is to cut the chess board horizontally above the Bishop, leaving him trapped in an area with few moves.

To be more specific: to win the Rook should try to reach H7 as soon as possible, if the Bishop is already in G7 the Rook should move to H5.

Proof:

If the Rook manages to reach row 7 before the Bishop, it automatically wins. Refer to the image below (I will mark in black the squares already moved across):

enter image description here

When the Rook is in H7 the most convenient position for the Bishop is to be in F6 and to have the next move. The possible movement for the Bishop are marked in yellow (the red marks indicate where he can't go to avoid being caught by the Rook on the Rook's next move), but both "yellow moves" leads to a situation where the Bishop is left with less moves than the rook.

  1. going to the upper yellow mark leads to 7 moves left for the Bishop and 24 moves left for the Rook
  2. going to the lower yellow mark leads to 6 moves left for the Bishop and 28 moves left for the Rook

To avoid the Rook going in H7, the Bishop should go in G7 on the first move, refer to the image below:

enter image description here

in this case the optimal move for the Rook is to go on H5, (he should not go in H7 to avoid being trapped with 0 moves left). Now the Bishop can only move in the yellow mark, this leads to 9 moves left for the Bishop and 20 moves left for the Rook.


Edit: In 2nd scenario the optimal move for the Rook is in H2, it leaves more moves than H5 (23 moves instead of 20). Thanks @Jason.

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  • 1
    $\begingroup$ In the 2nd scenario, why should the rook move to H5? H2 would be just as effective, while leaving more inhabitable squares, hence more optimal. If the bishop has already cut off half the board in the first move, so long as the rook stays on H the only valid move for the bishop is F8, and never the twain shall meet. The rook can putter around at 1 square a move the entire game. $\endgroup$ – Jason Aug 31 '15 at 20:12
  • $\begingroup$ @Jason you're right! I'll add it as a comment $\endgroup$ – Simone Sep 1 '15 at 12:29
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I think the Rook has the advantage in the first and third cases due to 2 reasons: the bishop is limited to a single color and the rook overall has more moves.

If the bishop starts with G7, then the rook could go H5, and then the bishop has to go F8 (H6 is deadly, as is H8). The bishop then has 10 moves left, while the rook has 23.

In general, for the first case, the rook wants to split the board so that bishop has minimal spaces, then just outlast it. Try to stay above the bishop, or if the bishop has a long diagonal, go to it to split the board.

For the third case, the same strategy as first case could work. If the rook goes E1, then the bishop moving 1-3 spaces means the rook can go E8 (E7 if dangerous), cutting the board in half. Anything other than that, E3, again cutting the board in half.

I don't really have a good idea for the second case, except that I would favor the Rook as it has more moves available to it. In fact, the Rook could use the "split the board" tactic, but only need to hit 4 squares.

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  • $\begingroup$ Yeah it looks like the Bishop can never win because it can't use the split the board tactic for any advantage, and both pieces limit its options twice as much as the rook. Any strategy, no matter how aggressive, never works in the favor of the Bishop. $\endgroup$ – Kingrames Aug 25 '15 at 13:19

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