No-Draw puzzles

All Things Chess
1

It was September 1, 2017, the day of the historic inaugural meeting of FIDNDC, an organization spun off by FIDE to launch an exciting new variant of chess.

Annie Australia: Ladies and Gentlemen, let us come to order. As you know, FIDE has empowered us to come up with a sensible set of rules regarding a new version of chess in which draws are not possible. They want to call it No-Draw Chess. As your chairperson, I am anxious to get this process started.

Bob Bosnia: I guess we need to consider each possible way a game of chess can presently end in a draw. For example, what about triple occurrence of position?

Charlie Canada: Yes, I’ve been giving that one some thought. I think we should simply make it illegal to make any move which would create a position which has appeared previously in the game (with the same player to move, of course). Never mind triple occurrence – we shouldn’t even allow a double occurrence!

Deborah Denmark: That’s a good idea. And if a player repeats a position, deliberately or inadvertently, the chess computer on which the game is being played will simply not allow it, and ask the player to take it back and make a legal move.

Ellen Estonia: What if the player has no legal move, other than one which repeats the position?

Fiona Finland: Then he has no legal move, period. He is either stalemated or checkmated, depending on whether he is in check.

George Germany: Speaking of stalemate, what do we do about that one? That can’t be a draw, either. Would it be a win for the stalemating player?

Annie: It has been said, of regular chess, that stalemate is the penalty for mauling without killing. If the stalemating player wins, we are removing that penalty. I think we should keep the penalty. The stalemating player should lose the game.

Bob: Wow! But I like it. A player with K+P vs K might be afraid to even try to win! :slight_smile:

Charlie: And what about K vs K? How do we make that not a draw?

Deborah. That’s an easy one. There are fewer than 4000 possible positions with K vs K – 8000, once you realize it could be either player’s move. Therefore, after at most 8000 moves by each player, there will be a forced repetition, and the game will be over.

Ellen: The same would be true of any normally “dead” position, I suppose.

Fiona: Yes – some positions are going to boil down to which player can eventually force the opponent to repeat a position. This opens up thousands of new possibilities!

George: True, but there is a problem. Didn’t we decide that the stalemating player loses? If you force your opponent to repeat, in essence you are stalemating him, forcing him to win.

Annie: Hmm, that’s true. I think we need to distinguish between two types of stalemate. A strong stalemate would be a position in which the player has no legal move, and would still have no legal move if repetitions were allowed. A weak stalemate would be a position in which the player has no legal move, but would have a legal move if repetitions were allowed.

Bob: So, what’s the difference, then? If you strongly stalemate your opponent, you lose, but if you weakly stalemate him, you win?

Annie: That’s what I was thinking, yes. And it certainly adds a new dimension to the game.

Charlie: I think we should apply the same concept to checkmate. In a strong checkmate, the opponent is in check and has no legal moves, nor any moves which would be legal but for repetition. In a weak checkmate, the opponent is in check and has no legal moves, but would have a legal move if repetitions were allowed.

Deborah: But, since we have declared checkmate and stalemate to be opposites (one wins, the other loses), the distinction between strong and weak should be opposite, too. If you strongly checkmate your opponent, you win, but if you weakly checkmate him, you lose.

Ellen: Perfect! Let’s see, what’s left?

Fiona: Of course, we disallow draws by agreement. Dead positions, we already took care of – eventually they will end in a forced repetition, game over. 50-move rule, ditto. As for draws by double time forfeit, we can simply set our computers to Halt At End mode, and have the computer declare the time forfeit as soon as it occurs.

George: And we don’t need to prohibit resignation, as that’s not a draw, either.

Annie: Then our work here is done. Meeting adjourned. (Bangs gavel.)

So, other than resignation or time forfeit, there are only four possible outcomes to a game of No-Draw chess:

Strong checkmate. Checkmating player wins.
Weak checkmate. Checkmating player loses.
Strong stalemate. Stalemating player loses.
Weak stalemate. Stalemating player wins.

Three No-Draw puzzles follow, in the next three posts.

Bill Smythe

2

Puzzle #1. See rules for No-Draw Chess in the top post. White to move and win:

Bill Smythe

3

Puzzle #2. See rules for No-Draw Chess in the top post. White to move and win:

Bill Smythe

4

Puzzle #3. See rules for No-Draw Chess in the top post. White to move and win:

Bill Smythe

5

Weak checkmate.

My head hurts, I can’t stand it. :open_mouth:

Puzzle 2:

  1. Be3 Ka8
  2. Kd8 Kb8
  3. Kd7 is a weak stalemate and wins.

Puzzle 3:

  1. Na6+ Ka8
  2. Nc7++ is a weak checkmate and loses!

but

  1. Nb5 Ka8
  2. Na7 Kb8
  3. Ng8 Ka8
  4. Ka6 Kb8
  5. Kb6 is a weak stalemate and wins.

Important: all solutions assume that none of the positions reached have ever occurred before. Given the board positions, that assumption could well be violated in game conditions.

6

Incorrect. 3. Kd7 is illegal, as it repeats the position which arose after 1. Be3.

Good point. Solvers, feel free to assume that any position you reach is not a repeat of any position which occurred before the initial position shown.

Bill Smythe

7

Wrong. 5. Kb6 is illegal, as it repeats the position which arose after 3. Ng8.

Keep in mind that the no-repetition rule in No-Draw chess is similar to the triple occurrence rule in regular chess. In both cases we are talking about a repeated position, not repeated moves. Different moves can lead to the same position.

Bill Smythe

8

Looks like the first two are winning weak stalemates:

  1. Ba6 Kb8 Bae2
  2. Bg3+ Ka8 Bf2
9

In the third one you have to avoid the losing weak checkmate of Nb5 Ka8 Nc7+. I don’t have time right now to look at it but I am going to guess that it involves black having to avoid a drawing but repeating move and thus getting checkmated.

10

3:

  1. Na8 Kxa8
  2. Ka6 Kb8
  3. Kb6 is a winning weak stalemate.

White gives the knight on his first move to avoid a repetition when he plays his third move. The position without the knight is not the same as the position with it.

In most cases (problems 1 and 2) one can simply be the first player to start in a nonchecking forced repetition sequence, and thereby execute a winning weak stalemate.

The difference in problem 3 is partly that the repetition involves check, leading to a losing weak checkmate. But even the nonchecking attempts don’t work, like my first try at the solution in my earlier post, because any move that does not lose a knight establishes a new position after white’s move that white would be the first to repeat; i.e. it loses the repetition-opposition. (This is partly the fault of knights: they cannot lose a tempo.) It would be OK if we could return to the original position, but that can only be done with check (lest Black’s king escape) and so the winning weak stalemate becomes a losing weak checkmate.

That was probably as clear as mud, but anyway …

Since in ordinary chess K+N+N vs. K I believe the weaker king can be forced into a corner, I think that a strategy like problem 3 could always be used to win this ending for the stronger side in Smythe’s No-Draw Chess.

11

These are nice examples where one gets the materialist’s “right” outcome in No-Draw where regular chess makes them draws.

The weakness of No-Draw is those positions (say, problem 3 without the knight on c7, indeed almost any ending we now consider a draw, problems 1-3 here being exceptions) where we don’t know who wins without running a whole cloud full of computers for hours or days to find the answer.

12

Phooey. You cooked my intended solutions:

  1. Bef3+ Kb8 Ba8 Kxa8 Ka6 Kb8 Kb6 weak stalemate.
  2. Bg3+ Ka8 Bb8 Kxb8 Kd8 Ka8 Kd7 weak stalemate.

Correct.

True. We must wait until computers can figure out all these positions instantly. When that happens, the players in an OTB No-Draw game could, instead of agreeing to a draw, agree to let the computer adjudicate the position, assuming best play by both players.

It seems virtually impossible to figure out who wins any K vs K position. Even on a 3x3 board (kings at a1 and c3) it’s difficult. (On a 3x2 board, however, it’s easy.)

Bill Smythe

13

Isn’t having the opposition a disadvantage in K v. K?

For example: White:Ka1, Black: Ka3 (edit: assume last move was a capture on a3)

White to move 1.Kb1 Kb3 … 7.Kh1 Kh3 8.Kg1 and now the Black king must retreat (8…Kg3 is illegal).

The choices are 8…Kh4 9.Kg2! or 8…Kg4 9.Kh2!

Lather, rinse, repeat.

14

I’m wrong (at least about the simplicity): 1.Kb1 Kb3 2.Kc1 Ka2! and Black will force White to take the opposition. 3.Kd1 Kb2 etc.

But 3.Kc2 Ka3 4.Kc3 (4.Kb1 is illegal) 4…Ka4 5.Kb2! and now I’m getting confused.

Or 3.Kc2 Ka1? 4.Kc1! wins.

15

That’s comforting.

And on an infinite x infinite board, K vs. K would always be a draw. (Many positions would be draws on such a board but some, like K+Q+Q vs. K, would not.)

So maybe we find out what positions of K vs. K if any are wins on a half-plane, then an infinite strip 8 squares wide, then an infinite half-strip 8 squares wide, and finally an 8x8 chessboard.

16

I think the first two cases (half-plane and infinite strip) are trivial draws in no-draw chess.

On an infinite half-strip eight squares wide (pace Frederick Jackson Turner, call the a-file boundary “West”) it’s obvious that the easternmost king can’t possibly lose, as it could run East to infinity. (Ditto for kings on same file.) But I don’t see how a defending king on a1–worst case scenario–can be prevented from escaping to the unbounded East, either. Using the opposition for containment would require position repetition.

The attacker can try to hip-check defender into the first or eighth rank, but the defender uses opposition to make attacker give way. The attacker gives way towards the long side (to try to squeeze defender on short side), but defender re-establishes opposition ad infinitum. There will be no repetition, as the attacker’s hip-check attempt always requires eastward movement.

If you cut and paste the following, stop at move 14, as move 15 is illegal on an 8x8 board.

[Event “No-Draw Chess?”]
[Site “Boundaries a1, a8, infinity 1, infinity 8”]
[White “Attacker”]
[Black “Defender”]
[Result “dead draw in no-draw chess”]
[SetUp “1”]
[FEN “8/8/8/8/8/8/8/k1K5 w - - 0 1”]
[PlyCount “30”]

  1. Kc2 Ka2 2. Kc3 Ka3 3. Kc4 Ka4 4. Kc5 Ka5 5. Kc6 Ka6 6. Kc7 Ka7 7. Kc8 Kb6 (
    7… Ka8 $2 8. Kc7 $18) 8. Kd7 Kc5 9. Ke6 Kd4 10. Kf5 Kd5 11. Kf4 Ke6 12. Kg5
    Ke5 13. Kg4 Kf6 14. Kh5 Kf5 15. Kh4 Kg6 15.Ki5 Kg5 etc.
17

There’s some similarity to go in the 8 x 8 case: ko fighting (obviously) and the ladder (in the limited sense that the kings’ moves in hipcheck situations may resemble the placement of stones in a ladder atari.)

18

How does it look like ko fighting? To do ko fighting you need at least 2 stones of each color, here there’s only one king of each color.

Yeah I agree it seems that the semi-infinite strip is a draw. Now then, how does it matter how long a finite strip is? If White wins in a 8 x n strip, how about an 8 x (n-1) strip, or how about an 8 x (n+1) strip? (I’m not sure which question is easier to answer. Without loss of generality, assume the kings are stationary with respect to the left “a” boundary.)

19

It may be a silly analogy (I’ve only reached “advanced kibitzer” at go), but moves of kings are roughly analogous to placement of stones.

20

I asked Prof. Noam Elkies about K vs. K in No-Draw Chess.