Repetition of moves: some dubious positions

[Ytreza]

Hello!

I tried to imagine some situations were the repetition rule as currently formulated would not be straightforward to apply. Let me know what you think about them, maybe I just missed something and it'd be the occasion to learn. I tried to think about plausible positions (not crazy imbricated forts )

The notation I use is the usual one, with the bottom of the board being the bottom of the picture.

First situation:
EDIT: remove the c11 pawn...

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White to play. White has no choice but to escape or the king is captured. But by doing so, he also threatens to reach the upper left corner. Then the only way for black to prevent white's win is to block. But by doing so, he threatens to capture the king. Then the only way for white to prevent black's win is to move the king. But by doing so, he also threatens to reach the open 10th row and win. Then the only way for black is to block. And so on.

In this case, who is the agressive player? Looks like it is a matter of point of view. Now, it is true that it is white who puts his piece in front of the path, and black who blocks, so white should lose according to the rules. But this is very odd: imagine now that the king is enclosed in the same way but in the middle of the board, with all corners sealed, so that white does not even threaten to win but to escape. In this second case, it is black who has a choice: he may win in another way than to capture right now. Conclusion: the side who has a choice is not necessarily the side who seeks the open path. And if we stick to the rule of figuring out who has the choice, in the second case (where the corner are sealed and white does not threaten a quick win) black needs to let white go, whereas in the first case he can't. Conclusion: white has better chances to win in the second case (as he loses in the first case). But white is clearly in a worse position in the second case, where all corners are sealed. Paradox?

Second situation:
EDIT: remove the a5 pawn

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Same idea, but more complicated because I wanted to show how such case could not necessarily involve the king. It is black's turn. Black needs to play f5 to b5 otherwise white captures c11 and wins with a kind of unusual guillotine. Then white plays b4 to f4, threatening a similar win on the other corner. So black needs to block, playing b5 to f5. Then white goes back to b4.

It looks like white has the choice here. But does it? If he lets black move the b5 pawn in b11, then black will be able to completely isolate the king, and white chances of win drop to zero: black will just use all of his time to capture the remaining pawns. So in this sense, it is actually black who looks for the path b5-b11 in order to secure the win, and white who blocks with b4-f4.

And what about black's choice? After all, all this requires some non straighforward reading. What if black d10-b10? after white b4-b11? What if black e3-c3 before the capture? Same thing on the right side. Black could have a lot of ideas (and actually I didn't read all variations so I'm not even sure of white victory even in the case he captures! but i'm pretty sure of black's victory if he immobilizes the king!) One can not even argue that white is clearly in a difficult position and should lose. He is very close to the corner with 2 supporting stones aiming at it! As black I would probably freak out, and be happy to have a way to secure a long win without thinking if the rules say that white must free the path.

Uncorrelated to the question of the repetition, I guess this second situation makes for a nice puzzle

Third situation:

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A more straighforward example to show that even such simple position can be dubious.
Black to play. The only way white can win is by threatening to go through the remaining open paths. Basically the repetition would be black j10-a10, white a9-k9, black a10-k10, white k9-a9 and so on. If white doesn't do that, black will be able to progressively close the remaining corner and then enclose the king. The only way to prevent this is to not give black the freedom to do it, by forcing blocking moves. Neither of the players have a choice there. White will surely lose if he doesn't repeat, black will surely lose if he doesn't block.

To summarize:
- Situation 1 shows that "having the choice" is not equivalent to "looking for the open path", and that the notion of "agressive player" is hard to define rigorously.
- Situation 2 shows that to figure out who has the choice is not straightforward: it requires to compute all the variations in order to compute whether a win is forced or not. It also shows that the player looking for the open path is not necessarily the "aggressive one" (it is black who tries to isolate the king and white who prevents him to do so).
- Situation 3 shows again that figuring out who has the choice is not straighforward even in a very simple situation.

Of course a way to make the rule rigorous is to stick to "the player looking for the open path must change his move". However this formulation is not really underlined in the rules, and is not fair to white as for white the game is all about looking for open paths while for black it is all about blocking (the first situation I showed is a great example of that!). In my opinion it is better then to say that white loses on repetitions (because white needs to put the king into safety and black only to prevent white from doing so).
Another possibility, more complicated, would be to say that white can follow up a repetition if it only involves the king on two squares, and in case black cannot enclose all the defenders white wins (which includes the edge fort). It could also be a draw (if not an edge fort, like in the first situation).

Hope you'll find these little problems interesting!
Ytreza

[Hagbard]

First situation:

White to play. White has no choice but to escape or the king is captured. But by doing so, he also threatens to reach the upper left corner. Then the only way for black to prevent white's win is to block. But by doing so, he threatens to capture the king. Then the only way for white to prevent black's win is to move the king. But by doing so, he also threatens to reach the open 10th row and win. Then the only way for black is to block. And so on.

In this case, who is the agressive player?

A brilliant example!!

By the way, the repetition rule has been reworded to:
"Perpetual repetitions are forbidden. The player who causes the repetition must find an alternative move or else he loses the game."
Exactly the same as before, but now expressing the spirit of the rule instead of details.

In your first situation,
the king is forced to move c6-b6, or else white loses immediately.
Black is then forced to move c7-b7, or else black loses immediately by the king next going b6-b11.
The king is then forced to move back b6-c6, or else white loses immediately.
Black is then forced to move back alco b7-c7, or else black loses immediately, this time by the king next going c6-c10.
and so etc. etc.

You succeeded in finding a case where the repetition is noone's fault, both sides are only doing forced moves, noone is driving this repetition.

This is so much a draw!

A machine (the software) might have great difficulty detecting such a sophisticated case; that is why we have a human umpire!

Update:
White captures c7 and wins.

[Hagbard]

Second situation:

And what about black's choice? After all, all this requires some non straighforward reading. What if black d10-b10? after white b4-b11?

Exactly. Black could let white capture the piece and then seal up the whole corner.
The battle is in full swing, and this game has not reached a point of repetetion.

[Ytreza]

By the way, the repetition rule has been reworded to:

I really don't like that a rule may be subject to interpretation. In the 3 cases I showed, it could lead to a debate with my opponent. For instance in the first situation, white loses in one move whereas black loses in 2 if white escapes on the left, and 3 if white escapes on the right. So white may refuse to see his victory and argue that it is black who has the choice...

The crucial point I want to make is that this repetition rule requires to know who has a forced win after the repetition cycle is broken and who has not. How I interpret the rule is that if one has a forced win, he must change his move. Now what you say is that if both players have a forced win, then it is a draw, whatever number of moves are needed to complete the win.

In this case, situation 2 and 3 should also be draws, even if it is less obvious that white loses if he breaks the repetition cycle.

Now consider the two following variations of the first situation:

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White to play. White has a choice! He can capture the b5 pawn and escape in the bottom. But... Black then encircles white in 3 moves. So is it a draw as in the previous case?

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Black to play. Black has a choice! He can block with the f10 pawn instead of the c7 pawn. But... White wins in >~ 50 moves thanks to a guillotine in the upper left corner. So is it a draw as in the previous case? And more importantly, should the players be able to compute more than 50 moves ahead in order to decide if it is a draw or a win for white?

And again, white can argue that he really has no choice, it is really obvious, he is captured in one move, whereas black seems to be able to block all attempt to escape... Think about two beginners for instance.

A machine (the software) might have great difficulty detecting such a sophisticated case; that is why we have a human umpire!

Actually machines are more capable of computing long variations than humans, so in order to apply the current rules in full generality we need a machine to solve hnefatafl and claim without doubt if one has a forced win or not after breaking a repetition.

[Ytreza]

Exactly. Black could let white capture the piece and then seal up the whole corner.
The battle is in full swing, and this game has not reached a point of repetetion.

What if you remove the b2 pawn and put a white pawn in f3? The point is that you can always adapt the showcase example in order to set a non straightforward forced win for white, that white can claim not to see such that black is forced to break the repetition. Same thing for black by the way: he could argue that white is far from having lost even if the king is isolated.

[Hagbard]

Third situation:

This is a classic. The eternal check. The first pattern of repetition which we met in the old days of Fetlar, it cropped up all the time.

White is driving this one, and if white keeps doing this repetition, then white loses by repetition.

Of course in this case, if white stops the repeating, he'll lose simply because he is in a losing position.

[Hagbard]

Now consider the two following variations of the first situation:

White to play. White has a choice! He can capture the b5 pawn

White is surrounded and has lost in a few moves.

[Ytreza]

White is surrounded and has lost in a few moves.

Only if he breaks the repetition, right?

[Hagbard]

Black to play. Black has a choice! He can block with the f10 pawn instead of the c7 pawn.

White wins this in a few moves.

[Hagbard]

The point is that you can always adapt the showcase example in order to set a non straightforward forced win for white, that white can claim not to see such that black is forced to break the repetition. Same thing for black by the way: he could argue that white is far from having lost even if the king is isolated.

I appreciate that your examples support your suggestion that all repetitions could be punished to white.

However, it is presumably possible to find examples where black is moving back and forth in repetitions attacking two white pieces. It would not be fair to leave white defenseless in this case, so that white was forced to stop blocking black and thus allowing black to capture one of the white pieces.