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Title: Reverse puzzles and impossible positions Post by leo on May 24th, 2010, 9:32am Has it been discussed before? The puzzle would be a weird position one has to find how it came to be. Incidentally, are there positions that are impossible to reach even with silly playing? For instance, a melee of interlaced freezings that can't be the result of any move? I can see one impossible position: 5 rabbits on goal. |
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Title: Re: Reverse puzzles and impossible positions Post by ocmiente on May 24th, 2010, 2:44pm Regarding impossible positions, I don't know if you could construct impossible positions that did not have something to do with a rabbit reaching the goal line. Although, maybe something having to do with immobilization might work. Including rabbits reaching goal, the one that came to mind was this: Code:
I don't think gold's rabbit could have reached that spot on one turn, and silvers rabbits could not have pushed it there. |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 2:55pm I wonder if there are any positions before the end of the game. I'll try to find one. Edit: just found one: http://lestrucsaleo.net/arimaa/boardimg.php?orient=n&size=300&imgtype=jpeg&ranks=///RRRRRRRR/rrrrrrrr/// I like this one because there are innumerable predecessor positions, but it could not have occurred in any game. |
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Title: Re: Reverse puzzles and impossible positions Post by rbarreira on May 24th, 2010, 3:04pm 722caasi that's not impossible I think, it's a EMHHDDCC handicap from both sides :P |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 3:06pm No, no, no, this is Gold's side of the board. either the gold line forms first, or the silver line does. Once one line has been made, the other one is impossible. |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 3:14pm I'm sorry, that position was possible, because gold pieces could have pushed silver rabbits into place, or vice versa. Here's a better one: http://lestrucsaleo.net/arimaa/boardimg.php?orient=n&size=300&imgtype=jpeg&ranks=//EMHHDDCC/RRRRRRRR/rrrrrrrr/emhhddcc// Still innumerable predecessors, but no pesky opposing pieces pushing rabbits around. |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 24th, 2010, 3:39pm Interesting thought, leo. I know that chess has yielded a reasonable body of "retrograde analysis" puzzles, but I assumed this was mostly to do with some quirky rules like en passant, castling, the prohibition against moving into check, and the obligation to move out of check. I would have guessed that Arimaa would have far fewer positions that are impossible or could only arise in one way, but after a couple of impossible positions by ocmiente and 722caasi, I have to wonder whether I am underestimating the possibilities of retrograde analysis for Arimaa. Perhaps jammed up positions with a lot of freezing restrict the move options enough that one could work backwards. |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 5:20pm Here's a nice problem: What did gold just play: http://lestrucsaleo.net/arimaa/boardimg.php?orient=n&size=300&imgtype=jpeg&ranks=3Re//3c///// There are only two possibilities. Try to find both. |
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Title: Re: Reverse puzzles and impossible positions Post by FireBorn on May 24th, 2010, 5:27pm The rabbit could've started on any of a7, b7, c7, or b6 and gotten to that square, right? |
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Title: Re: Reverse puzzles and impossible positions Post by Sconibulus on May 24th, 2010, 5:29pm Or any number of positions involving unfreezing the rabbit on d7 and piece sacrifice |
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Title: Re: Reverse puzzles and impossible positions Post by FireBorn on May 24th, 2010, 5:33pm I don't think this one's possible: Code:
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 7:31pm Remember, FireBorn and Sconibulus, the game ends if one side has no rabbits at the end of a turn. Neither of your ideas work, as stated. |
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Title: Re: Reverse puzzles and impossible positions Post by ocmiente on May 24th, 2010, 7:46pm That's a good puzzle. Wouldn't have caught that for a long time without the clue. I think one solution is: Rd7N re6E Ee7S Ee6E <EDIT> And the other must be: rc7S Eb7e Rd7N Ec7s But in this case, you don't have to use an elephant to make this work, and the elephant could start from c8 too. |
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Title: Re: Reverse puzzles and impossible positions Post by The_Jeh on May 24th, 2010, 7:59pm Ed6w ce6w rf6x Rc7n Ec6x Rc8e |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 8:14pm I'm sorry for the inaccuracies. I believe this puzzle fixes all the problems and limits it to one solution. http://lestrucsaleo.net/arimaa/boardimg.php?orient=n&size=300&imgtype=jpeg&ranks=3R/2c1e////// Is the only solution Rd7n re6e Ed6e Ee6e? |
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Title: Re: Reverse puzzles and impossible positions Post by ocmiente on May 24th, 2010, 8:18pm No, I don't think so. There is another... and another... (just two main others that I can think of. Try starting the elephant at e6 or something else at d5 <EDIT> Oh... wait.. forgot about that cat... just a minute... <EDIT> I think you're right... |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 24th, 2010, 8:22pm Yeah, that cat is vital. I still think it's unique. |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 25th, 2010, 6:58am on 05/24/10 at 17:33:22, FireBorn wrote:
It looks possible with a c-file or f-file rabbit having been pushed into position to complete the two rabbit lines, and then the piece(s) that enabled the push committing suicide. <EDIT> Oh, maybe not, because there must have been a friendly piece keeping the rabbit safe on the trap before the push, and then there is no place for that piece to suicide later. Subtle point! |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 25th, 2010, 7:12am on 05/24/10 at 20:22:29, 722caasi wrote:
I, too, can't find a second solution. Nice puzzle, thanks! But Silver could also have bought about the terminal position in many ways, so technically Gold's last move could have been many, many different things if it was Silver who ended the game. I guess the question, "What did Gold just play?" implies that Gold "just" played, meaning Silver hasn't any time to play since then, but to leave no room for pedantic quibbling, one might instead frame the question as, "Gold made this game's last move. What was it?" |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 25th, 2010, 11:26am I have counted The last puzzle has the only solution, switched all pieces cannot be achieved and two switched rows of rabbits except the f,c ones as well. |
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Title: Re: Reverse puzzles and impossible positions Post by FireBorn on May 25th, 2010, 12:54pm I don't think this one's possible either: Code:
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Title: Re: Reverse puzzles and impossible positions Post by FireBorn on May 25th, 2010, 1:02pm Neither is this one: Code:
This one is possible: Code:
But this one is not: Code:
Nor this one: Code:
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Title: Re: Reverse puzzles and impossible positions Post by FireBorn on May 25th, 2010, 1:25pm I'm trying to come up with one involving immobilization but for the life of me I can't |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 26th, 2010, 8:50am Twin puzzles:
Would the solution differ for the other players turn? |
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Title: Re: Reverse puzzles and impossible positions Post by leo on May 26th, 2010, 9:17am You guys sure are inventive! :D Hippo: that last one of yours is most strange, whose turn is it? EDIT: OK, you updated your post. I'll try to solve the puzzles. FireBorn: yeah, immobilizations and forced moves should give some good puzzles Fritzlein: retrograde analysis, will remember the term ^^ 722caasi: because of my upcoming web domain change the board picture urls will soon become invalid. you can either save them and upload them on imageshack or such -- or change the urls when the transfer is done: see this thread (http://arimaa.com/arimaa/forum/cgi/YaBB.cgi?board=talk;action=display;num=1273874607) |
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Title: Re: Reverse puzzles and impossible positions Post by FireBorn on May 26th, 2010, 12:46pm @Hippo: My understanding of the repeat-rules is fuzzy, but I think in the first one Silver is winning because he can shuffle his f8 rabbit back and forth while Gold has to keep coming up with new board positions, meaning his h6 rabbit will eventually have to move to g6, freeing Silver's cat, and once it's free Silver can block off that corner of the board with his rabbits and decimate Gold's rabbits with his cat. Similar situation if it's Gold to move, except silver will be forced to move a rabbit out of the way, allowing Gold to goal next move. |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 26th, 2010, 2:24pm Try to show possible last moves preceding the position. They are more natural for the case Silver is winning. And a bit confusing for the case Gold is winning. |
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Title: Re: Reverse puzzles and impossible positions Post by 722caasi on May 26th, 2010, 2:34pm Hippo: The last move could just have been a suicide. What does that imply? |
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Title: Re: Reverse puzzles and impossible positions Post by RonWeasley on May 26th, 2010, 3:27pm These are great! |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 26th, 2010, 10:20pm on 05/26/10 at 14:34:08, 722caasi wrote:
OK, nice solution to both problems as they are stated. May be explaination of the winning strategy would help :). |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 27th, 2010, 4:32pm Hippo, I am not sure I understand your puzzle. You might mean to say this: "From the given position and the information that it is Gold's move, you can deduce that Gold is winning. Explain this deduction." But that conclusion doesn't follow, does it? So instead I am guessing that you mean to say this: "From the given position and the information that it is Gold's move and the information that Gold is winning, you can deduce exactly why Gold is winning." In this case, the only reason that Gold could be winning is that the repetition rule will force Silver to sacrifice a rabbit or move away the cat from blocking goal. Is there something more specific that we should be able to deduce in addition? It seems that, depending on prior moves, Gold might have more than one "winning strategy". For example, it might be that the move Ra6e forces Silver to immediately sacrifice a rabbit because other moves all result in 3-fold repetition or immediate goal. Or it might be that this move loses while Ra6e Rb6e wins. |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 27th, 2010, 10:48pm on 05/27/10 at 16:32:16, Fritzlein wrote:
This is among retrograde analysis. Yes, it has a lot of solutions so it is not correct retrograde puzzle. Yes you must know the game history to decide who wins, who's turn it is is not important. 722cassis statement ... last turn move describtion ends with x ... the player to move wins. Is the shortest describtion of sufficient history information. So the current version of puzzle says: If the last move ended with x. Show the strategy of player to move leading to victory. (Try to find as easy describtion as possible). ------------- Oops I was trying to make some other puzzle and it is not correct (neither version) ... how could I delete it from the collection? I have not linked them yet to any page. But following one is OK: http://arimaa.com/arimaa/puzzles/show.cgi?p=p55. I have forgot how to link it to the puzzle page. ... (goal in more than 15 ... exact value is ..). |
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Title: Re: Reverse puzzles and impossible positions Post by jdb on May 28th, 2010, 6:10am Nice puzzles Hippo. Silver can suicide a rabbit and still maintain the blockade. I think this means silver wins with either side to move. |
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Title: Re: Reverse puzzles and impossible positions Post by Manuel on May 28th, 2010, 7:23am on 05/28/10 at 06:10:01, jdb wrote:
I don't think silver can suicide a rabbit and keep maintaining the blockade, is it? Silver has to do some move in every next turn... |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 28th, 2010, 7:50am on 05/28/10 at 06:10:01, jdb wrote:
Yes, it looked like that to me too at first, but after suiciding a rabbit Silver no longer has an extra piece to shuffle, and thus gets hit by the no-pass rule. on 05/27/10 at 22:48:55, Hippo wrote:
So we are allowed to give only the last move that occurred before the position, and that addition by itself must be sufficient to prove that the current player on move is winning? I see how to do this for Silver to move, but not for Gold to move in the same position. |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 28th, 2010, 9:06am :) I have trained a bit to win on 3 repetitions :). The strategy for Gold is the same as for Silver :). Seems study of this is not well known so far. P.S.: The only important thing is gold and silver plays on independent playgrounds. Sacrifying any number of pieces would not change the result. |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 28th, 2010, 10:27am on 05/28/10 at 09:06:30, Hippo wrote:
I believe you that you understand this type of position better than any of us. Good job! Thanks for the puzzle! But would you be so kind as to clarify exactly what the puzzle is asking? I don't want to think I have solved it (or busted it) and then have you tell me you meant something else. Are you now saying the second puzzle consists of explaining how Gold can win given that Silver moved last and that move was a capture? |
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Title: Re: Reverse puzzles and impossible positions Post by jdb on May 28th, 2010, 10:51am on 05/28/10 at 07:50:56, Fritzlein wrote:
The shuffle is still possible, it just has to involve at least two pieces. |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 28th, 2010, 11:24am on 05/28/10 at 10:51:21, jdb wrote:
No, because if e7 or d7 is left unoccupied by Silver, then Gold can occupy it, curtailing Silver's shuffle strategy. Only c7 and f7 can be left unoccupied, but Silver would need six steps to swap between those two. |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 28th, 2010, 12:02pm on 05/28/10 at 10:27:47, Fritzlein wrote:
OK, I was asking how to apply the advantage caused by your opponent moving to the position when players play on separate playgrounds. I surely enjoy to explain some thing about arimaa to you:) ... but may be it's rather some abstract graph theory game? Let denote such positions by two letters ... lower case for our position and upper case for opponent's one. We start at position a* and move to position b*. We move only between a* and b*. Why we cannot lose on 3 times repetition in position xY? Because opponent plays to zY third time turn before us. ({x,z}={a,b}). Induction ... we play to xX k-th time move after opponent payed to zX k-th time... |
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Title: Re: Reverse puzzles and impossible positions Post by Fritzlein on May 28th, 2010, 12:29pm Er, I was still trying to clarify the question, but thanks anyway for giving me the answer, which indirectly tells me that my last guess at the question was in fact correct. :) I had a strong intuition that the player with more squares to shuffle between would always win, but I had just seen that was not the case for two versus three, so I was just starting to understand that as long as I have two squares to shuffle between, it doesn't matter how many squares my opponent has, or in general how many positions my opponent can reach. I might have gotten all the way to the correct answer in a few more iterations. Potentially you have just provided us with a way to resolve some practical draws, i.e. games in which both players can't or won't do anything but shuffle pieces. For example, in the position below, assuming Silver just completed the rabbit wall, we could allow Gold to claim a win by declaring an intention to play Ea1n and Ea2s on alternate turns from here on out. Although Silver could avoid repetition for many lifetimes, Hippo's Theorem proves it is theoretically won for Gold. http://lestrucsaleo.net/arimaa/boardimg.php?orient=n&size=250&imgtype=jpeg&ranks=emhhddcc///rrrrrrrr/RRRRRRRR///EMHHDDCC Quick, upload the puzzle "Gold to move and win in seven trillion". :) Note this is different from the current rules which would require the game to be played until time was exhausted and then award the win to Silver based on the game having been captureless. Depending on how and whether such a rule could be applied in practice, it would be an attractive feature that whoever creates a divided-board situation is theoretically the loser. Thus both players have an incentive not to do it, and instead keep open a channel for engagement. Even if there is no good way to apply the theory in general, it could practically decide some games even within the time limit. For example in Arimaa World League game Nevermind vs. Sconibulus (http://arimaa.com/arimaa/games/jsShowGame.cgi?gid=143706&s=w), it appeared around move 21 that if Nevermind wasn't careful, Sconibulus could complete a swarm to lock up the board and create a piece-shuffling contest. But even if Sconibulus has created such a deadlock, perhaps Nevermind, armed with the knowledge from this thread, could have forced Sconibulus to deviate even within the game time controls by simply alternating Rh1w and Rg1e. Thanks for teaching me something about Arimaa! (But of course you don't need a puzzle to do that; you are teaching me something about Arimaa simply by beating me in our Postal Mixer game.) |
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Title: Re: Reverse puzzles and impossible positions Post by Hippo on May 28th, 2010, 1:12pm on 05/28/10 at 12:29:23, Fritzlein wrote:
Sorry about that (my poor English :)) on 05/28/10 at 12:29:23, Fritzlein wrote:
Yes, it would be nice to be able to declare victory on 3-times repetition based on my theorem :). The player who divides the board to independent playgrounds loses. (If the opponent declares this kind of victory (and has at least 2 positions)) ... (rabbit step forward creates another playground as it is irreversible by rabbit's owner, as well as removing a piece ... own or opponent's). http://lestrucsaleo.net/arimaa/boardimg.php?orient=n&size=250&imgtype=jpeg&ranks=/1e2C1C/1M1Dh1hD/rrrrrrrr/RRRRRRRR/1m1dH1Hd/1E2c1c/ In this position taking a piece is losing move (the player completting the wall was losing after that move, but he is probably not lost after inaccurate opponent's play). on 05/28/10 at 12:29:23, Fritzlein wrote:
It's very far to say that :) Adanac is used to give a horse to win afterwards against much better player than I am, you are better than him and you gave me much less ... It's probable it's just a matter of time a blunder of me would decide the game ... but surelly I would try to avoid that. |
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