View Full Version : K+R vs. K

Czentovic

17-01-2013, 08:32 PM

Consider wKa1,Rb1/bKc2 (or FEN:8/8/8/8/8/8/2k5/KR6 w) on a double chessboard (files A to P and ranks 1 to 16). White wins by checkmating Black, Black wins by making 50 moves or reaching the far edges of the board (rank 16 or file P)

[1] With best play, who wins? (When I gave this problem to a group of juniors to whom I was teaching the K+R vs. K ending, Black won nearly all the time, usually by reaching file P)

[2] What is the smallest rectangular board on which White can force a win?

Garrett

18-01-2013, 08:44 AM

White definately wins, not sure just at the moment about the smallest board bit, will try to work it out after work....

cheers

Garrett.

Kevin Bonham

18-01-2013, 11:59 AM

Interesting puzzle. My suggestion in white (hope I'm wrong):

I'm proposing black wins on a board of any size and even without the 50 move rule by the following method:

1. If possible move the king diagonally towards the far corner (ie increase both file and rank by one).

2. If this isn't possible, it's because the rook is on either the next file or rank. In this case move king towards whichever far side can be moved towards.

Note that the king can always do either 1 or 2 unless the rook is on both the next file and the next rank (ie diagonally adjacent) and in this case the king can take the rook unless it is protected by the other king.

It follows that black must eventually win, unless it is possible for white to get his king "around" the other king to protect the rook in at least one direction.

Doing so would require white to, at least, have moved two more squares than black in one direction and as many squares as black in the other.

But given that:

(i) Black starts two squares closer in one direction, even assuming white moves first.

(ii) White must make a bare minimum of two rook moves to stop black winning, each of which allows black to increase his lead in one direction, the other, or both, compared to white's king.

(iii) Black's king often obstructs white's king from getting around it if white tries to make progress by moving diagonally while black's king is unable to.

...I just don't see how white can do it.

I'd be very happy to be proven wrong on this as I think the winning method for the other side would be very interesting if there was one.

Garrett

18-01-2013, 12:20 PM

I should probably explain

White plays Re1-e lots

as Black is marching the king up the d-line white plays Ka1-b1 then diagonally towards top right corner.

When the black king reaches the rook the rook plays to the P-line.

As the black king marches along the 14th or 15th rank he will encounter the white king in front of him.

Then the standard procedure to force the Black king back.

Garrett

18-01-2013, 01:03 PM

I am going to stab a guess that a 9x9 board is the smallest square board to force this win.

This means the rook only has to go to the 9th rank on larger boards.

cheers

Garrett.

Kevin Bonham

18-01-2013, 01:32 PM

Yep. My assumption (2) was fatally flawed. Just checking if 9x9 works. If it does it shouldn't be hard to get mate in 50.

Kevin Bonham

18-01-2013, 01:38 PM

I get 10x10 not 9x9

9x9:

1.Re1 Kd3

2.Re9 Kd4

3.Kb1 Kd5

4.Kc2 Kd6

5.Kd3 Kd7

6.Ke4 Kd8

7.Ri9 Ke8

8.Kf5 Kf8

10x10:

1.Re1 Kd3

2.Re10 Kd4

3.Kb1 Kd5

4.Kc2 Kd6

5.Kd3 Kd7

6.Ke4 Kd8

7.Kf5 Kd9

8.Rj10 Ke9

9.Kg6 Kf9

10.Kh7 Kg9

11.Ki8 and Kh9 is impossible.

Garrett

18-01-2013, 02:09 PM

okay, well Czentovic did say rectangular

so it seems the best we have so far is 10 ranks and 9 files.....

cheers

Garrett.

Garrett

19-01-2013, 03:11 AM

There could be more to this.

Black can probably try playing Kc2-d2-c1 to get the opposition.

White might have to play Ka1-a2-a3 then make a waiting move with the rook to force black to play Kc4 then White can outflank with Kb2 etc.

This might require a slightly larger board, because if White plays his rook to the last rank straight away then the waiting move will place it on a less than ideal square. If White plays rook to second last rank then Black might be able to forget about the opposition thingy and go harrass the rook straight away....

Czentovic

19-01-2013, 07:49 PM

There could be more to this.

Black can probably try playing Kc2-d2-c1 to get the opposition.

White might have to play Ka1-a2-a3 then make a waiting move with the rook to force black to play Kc4 then White can outflank with Kb2 etc.

This might require a slightly larger board, because if White plays his rook to the last rank straight away then the waiting move will place it on a less than ideal square. If White plays rook to second last rank then Black might be able to forget about the opposition thingy and go harrass the rook straight away....

Yes, there is! I spent some time choosing the initial position of the Rook and Kings so that you need to find a way for White to avoid opposition by Black.

Also, White can mate on a rectangular board with less than 90 squares.

Zwischenzug

22-01-2013, 04:41 PM

Is this the position? But with twice the size on the right?

8/8/8/8/8/8/2k5/KR6 w - - 0 1

Kevin Bonham

22-01-2013, 04:50 PM

Twice the size in both directions, ie four chessboards.

Zwischenzug

22-01-2013, 05:50 PM

So this is the position?

2167

Kevin Bonham

22-01-2013, 05:55 PM

As I understand it, yes.

Czentovic

23-01-2013, 12:21 PM

Time for a hint: the smallest rectangular board on which White can force a win is 8 by 10 - find the win.

Czentovic

25-01-2013, 09:56 AM

Here's the answer:

1. Rb4 Kd3

2. Kb2 Ke3

3. Rj4 Kf3

4. Kc3 Kg3

5. Kd4 Kh3

6. Ke5 Ki3

7. Rj8 Ki4

8. Kf6 Ki5

9. Kg7 Ki6

10. Kh8

If Black tries for the opposition with Kc3 instead of Kd3 then

1. Rb4 Kc3

2. Ri4 Kb3

3. Kb1 Kc3

4. Ka2 and White wins on a smaller board.

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