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Subject: Do 2 Queens & 1 Drone lead to a draw? rss

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Russ Williams
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Is there any established endgame theory about Martian Chess? We had what seemed to be a draw situation today, with 2 Queens on a2na's side and a Drone on my side. The Queens could never simply catch the Drone (it can always move sideways to a different unthreatened column) so a2na moved 1 Queen to my side to try shaking things up.

We continued to maneuver for a while and eventually agreed to call the game a draw. Our scores were equal at that point - if one of us had been ahead in points, I guess the situation would become more problematical/confusing rules-wise!

The rules printed in Playing With Pyramids seem to have no mention of draws or how to resolve a seemingly endless situation. I have no idea if people typically play with some kind of rule that it's illegal to repeat a previous position, or that repeating a previous position automatically ends the game in a draw, or what.

When we first learned the game, we thought draws would be a common problem, but in fact they don't seem to be. It's a cool game we're both enjoying a lot.
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Ax Bits
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I think I've solved the draw. It takes advantage of the "no givebacks" clause for the 2 player game.

Given 2 Qs, arrange them like such:
....
....
....
Q..Q
----
----
-dd-
-dd-


The drone must be occupying one of the 4 d spaces to avoid immediate capture. It doesn't matter which one but we'll assume one for concreteness.

Player Top moves like so:
....
....
....
Q...
----
-Q--
----
-D--

so as to place the Q in the same column as the D.

Player Bottom must move a piece across the canal (Top will simpley dump her remaining Q otherwise and end the game). Bottom then must return his new Q to top. He may not return it whence it came and is forced to place it one any one of q

.q..
.q..
.q..
Qq..
----
-Q--
----
-D--


Top will then respond by capturing D.

So Bottom can not return the Q, lest he lose the D and give Top 2 points. He keeps the Q and leaves Top unable to capture, but she moves her Q across and the game ends.
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Russ Williams
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That analysis makes sense to me (assuming one player can force the type of arrangement you start with), although then the game ends in a tie (in my scenario, we had equal points). Still, of course a tie is better than a drawn out non-terminating game. And if one player had more points, perhaps in this QQD scenario it would possible for them to force the game to terminate then, assuming they can get the Q..Q etc situation you present.

A similar harder situation arose a few days ago:

Instead of QQD, we had 3 queens in the end, and 1 player had more points. It seemed tough for the player with more points to actually end the game. Every time they moved 1 of their 2 queens across, the losing player (with fewer points) could move 1 of their 2 queens back. (2 queens on your side gives more possibilities to send a piece back than 1 queen and 1 drone...) I wonder if there is a way to set up a similar situation with QQQ as you did with QQD.
 
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Ax Bits
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I believe the situation I drew would work with a Q in place of the D.

I'll use 1 2 and 3 for the 3 Qs:
....
....
....
1..2
----
----
----
-3--

then Top makes her move the same as previous
....
....
....
1...
----
-2--
----
-3--

Bottom is stuck. If he returns 2, Top will use 2 to capture 3. 3 can not cross, 2 is blocking the only route. If neither crosses, Top moves 1 across on her turn and the game ends.

Let me figure out if there is a way to deny the initial state.
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Russ Williams
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Maxx_Pointy wrote:
Let me figure out if there is a way to deny the initial state.

That seems the key. It seems like it would be easy to prevent North from achieving the necessary initial state to pull this off.

BTW, do you play at superdupergames.org, and if so are you up for a game?
 
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So, it seems that Bottom can deny Top a chance to form the initial state by continuously threatening one of the two spots (Top can easily claim the other one). It's only possible to do so with a queen.

As such I think the 2Q vs 1Q may be unending.


I don't play there, but I'll check it out.
 
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Joshua Kronengold
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FWIW, I've gotten into the "three queens" situation a lot on SDG, and generally found that I could force things into a winnable situation every time. So I simply don't think it's non-terminating.

Let's examine the setup for a moment. Assume that the leading player has two queens and can get one into one of the key positions, so:
A B C D
x x x x 1
x x x x 2
x Q x x 3
Q x x x 4
=======
x x Q x
x x x x
x x x x
x x x x

B3-D3 | C5-B6
D3-C4

At this point, the only move Bottom has that defends D4 is B6-D8. But that's a loss -- White responds with A4-D7, forcing the same endgame without needing to have both pieces in position.

I'm frankly not sure the game has any true draw positions that aren't either 1 piece on one piece or positions with tied points. (rather than infinite loops with uneven points).

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mneme wrote:
At this point, the only move Bottom has that defends D4 is B6-D8. But that's a loss -- White responds with A4-D7, forcing the same endgame without needing to have both pieces in position.


B6-D6 instead.
 
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Russ Williams
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mneme wrote:
FWIW, I've gotten into the "three queens" situation a lot on SDG, and generally found that I could force things into a winnable situation every time. So I simply don't think it's non-terminating.

Thanks for sharing your (certainly more advanced than mine) experience/knowledge!

Quote:
I'm frankly not sure the game has any true draw positions that aren't either 1 piece on one piece or positions with tied points. (rather than infinite loops with uneven points).


Cool; so next time we play, if I find myself behind I won't feel cheesy and annoying if I try for a draw.

I am confused why, in your example, after top's B3-D3:
A B C D
x x x x 1
x x x x 2
x x x Q 3
Q x x x 4
x x Q x
x x x x
x x x x
x x x x

the bottom player must (?) reply C5-B6; it seems like they could go to C7 or C8 also. I guess you're implicitly saying B6 is their strongest response, probably because retreating further back will be worse?

I wish some enterprising bored programmer would exhaustively analyze various endgame positions...
 
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russ wrote:
I am confused why, in your example, after top's B3-D3:
A B C D
x x x x 1
x x x x 2
x x x Q 3
Q x x x 4
x x Q x
x x x x
x x x x
x x x x

the bottom player must (?) reply C5-B6; it seems like they could go to C7 or C8 also. I guess you're implicitly saying B6 is their strongest response, probably because retreating further back will be worse?


Bottom wants to maintain the threat at D4, the only way to do that while remaining unthreatened is C5-B6.
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Russ Williams
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BTW:

The official 2-player rules say you can't move a just-crossed piece right back across to the square it started at, but you can move it to any other square (on either side of the board). Playing at superdupergames lately, it seems like SDG over-strictly prevents you returning a just-crossed piece back to the other side anywhere. Has anyone else noticed that, or am I just hallucinating?
 
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