Is it possible to draw a game in Alveole?

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Dr C
Posts: 5
Joined: 14 December 2018, 12:15

Is it possible to draw a game in Alveole?

Post by Dr C »

Here is an interesting theoretical question:
Knowing that if a player can't move or forces their opponent down to a single piece loses, I was thinking that a draw could still be possible.
The idea is that there may be some positions where both players are close to winning. Supposing that at each turn the players find themselves in a situation where only a single avoids them to lose and simultaneously forces the opponent in a similar situation. If the initial position is repeated, this would lead to a infinite cycle of moves. And therefore no one would be able to force a victory.

To be clear this would be analogous to a chess position with white to move:
White's King is in e4, and they have two pawns in c4 and a4.
Black's King is in e6, and they have two pawns, opposing the white ones, in c5 and a5.
The only move that avoid losing is moving the white king to f4. The only move that allows black to not to lose now is King to f6, forcing white to play the King back to e4 and consequently forcing black to do the same playing King to e6. This would lead to an infinite loop, which in chess is stopped by the triple repetition rule that woukld end the game in a draw.

Therefore what would happen in Alveole in this case? But first is it even possible that such situation exists?
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Romain672
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Joined: 05 April 2016, 13:53

Re: Is it possible to draw a game in Alveole?

Post by Romain672 »

The joined situation is (edit: look like) a draw. But you obviously can't reach it in a normal game.

edit²: added a second situation (top), normally much more convincing. But it's still not clear if it's a draw or not.
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Dr C
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Joined: 14 December 2018, 12:15

Re: Is it possible to draw a game in Alveole?

Post by Dr C »

@Romain 672.. I don't think they are theoretical draws as that chess position.. in the second case whoever moves can capture the opponent's piece in the center of the board and then win..
While in the first one I think that anyone who would attack first would be able to take the lead on the game..
So not positions that can lead to an infinite loop..
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Dr C
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Joined: 14 December 2018, 12:15

Re: Is it possible to draw a game in Alveole?

Post by Dr C »

I believe the position in the ASCII strings below should be draw! Unfortunately BGA did not let me upload a picture. For convention the board is rotated by 30°, RED stones are R1,R2,R3 and blue's B1,B2,B3.

After some analysis, I've arrived at the following conclusion: assuming it's RED's turn to play (for convention, as BLUE to play is equivalent). RED cannot move pegs R2 or R3, as doing so would result in a loss in at most 2 moves. Therefore RED must move peg R1. If peg #1 is moved to "a", it follows that BLUE must move peg B4 to position "e". RED is then forced to move peg R2 to position F, and BLUE must move peg B6 to position "g". This results in the same position, shifted one line down. If peg R1 is moved to position "b", then BLUE is forced to move peg B5 to "h", resulting in the same position but reflected along its horizontal axis. Moves to positions "c" or "d" eventually result in a loss for RED (albeit through very long lines of play). This seems to hold true even if the position is shifted up or down, even as much as when some of the pegs are in contact with the border of the board (leading to equally longer sequences of moves).

Summarizing: RED must play peg R1 to either position "a" or "b" in order to avoid losing, and BLUE is forced to make moves that restore the position but shifted u or down, or reflected; leading to an endless cycle. Thus, the only solution in this position would be for both players to agree to a draw.


*Board rotated by 30 degrees clockwise.
█████████████████████████████████████████
█████████████████████████████████████████
██████████░░░█░░░█░░░█░░░█░░░████████████
████████░░░█░░░█░░░█░░░█░░░█░░░██████████
██████░░░█. c ..█░░░█░░░█░░░█░░░█░░░████████
████░░░█░░░█░░░█. b . █. h . █░░░█░░░█░░░██████
██. d . █░░░█░░░█ -R1-█-B6-█ -R2-█░░░█░░░█░░░████
████░░░█░░░█. a ..█ -B5-█-R3-█ -B4-█░░░█░░░██████
██████░░░█░░░█. g ..█. f . █. e ..█░░░█░░░████████
████████░░░█░░░█░░░█░░░█░░░█░░░██████████
██████████░░░█░░░█░░░█░░░█░░░████████████
█████████████████████████████████████████
█████████████████████████████████████████


It's very likely that there are many more such positions with similar characteristics. Would be interesting to see other ones.
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