AbigailII: I believe that proves an infinite game, not an infinite number of games. Small but important distinction.

Pbarb2: If you see "No days off" written in red in a game, the auto-vacation will not prevent its timing out. See this:
http://brainking.com/cz/Help?ht=13

I guess that the no days off makes the difference and makes the automatic vacation thing useless..........

Pedro Martínez: I see that Pedro.. But why did I run out of time with my automatic vacation on?

Pbarb2: Tournament: The Delete's Gammon Fest #1
Game ID: 890904
Score of finished games (Pbarb2 - MsDelete, Backgammon Race): 0 : 1 (= 0) (show games)
Time per move (?): 3 days, no days off
Public game (visible for other players)
Rated game (the result will be calculated for players' BKR)
Board size: 1 (change)
Layout: columns (change)
White ran out of time.
Black is the winner.

I just got this in my message box that I timed out in this game. I have my automatic vacation box checked so it is in use. Just wonder WHY?
Tournament: The Delete's Gammon Fest #1
Game ID: 890904

grenv: One piece all works as well. Put them anywhere on the board, under the condition they still have contact. Now let them roll only 1-1s. Neither side will be able to bear off.

Pedro Martínez: lol...indeed ..but nobody24 and me are friends and we often play together on the same computer ..That's why we just noticed the difference :)

playBunny: The simple case is always the best place to start, and I agree it was a clever proof.

One piece each wouldn't work, it relied on players hitting each other's block.

Peeky: Yes, but you can see only how your rating will change in case of win, draw or loss. You cannot see the opponent's rating chnages...

You can see it underneath the game board ....in every game there's mentionned how your rating will change in case of win, draw or loss

Peeky: How did you learn how many points would nobody24 lose?

Just noticed something strange . I'm playing a normal BG game against nobody24 .... he has a rating of 2134 and I have a rating of 1805. When I loose the game my rating will go down with 30 points . When he looses the game his rating will go down with 11 points though his rating is much higher than mine .I thought it would be opposite because my rating is so low. Can somebody explain this to me ?

playBunny: In other words, if you make a roll, the probability that something will appear on the dice is 1. The probability of any specific roll or sequence of rolls is lower than 1, no matter if you consider the number of moves finite or infinite.

Pedro Martínez: Aye, and the "something" in question is the endless sequence of double fives.

Or did I miss something? I wasn't sure whether you were agreeing or challenging?

Perhaps by the probability of "something" = 1 you mean the probability of any of the sequences implying that the probability of a particular sequence is less than 1? As I understand it, that's true when considering the finite but not when considering the infinite.

grenv: "the simple case with 2 pieces each left." Lol, I think that's actually quite crafty.

But why complicate it? Let's use the case of only 1 piece each. ;-)

playBunny: The probability of rolling "something" is 1, i.e. 100%.

Chessmaster1000: Excuse my ignorance, I'm a logician more than a mathematician, but I would have thought that the probability of an endless sequence of 5-5s is exactly 1.

Consider every single possible infinitely long sequence of dice rolls.
Surely 5-5, 5-5, 5-5... is among them? If not, why not?

Well done everybody, at the start of the thread I thought I had a problem to solve, but it was already solved by the time I woke up. There are indeed an infinite number of games, as proved by the simple case with 2 pieces each left.

As for the infinitely long game, I agree with Wil. it is possible in theory to have an infintely long game. The idea that it's probability approaches zero only shows us that we have no possibility of ever seeing the game to completion. This, of course, is the point.

Chessmaster1000:
For every move it's 1/36.
For 2 moves to happen is (1/36)^2
For 3 moves to happen is (1/36)^3
For an infinite number of times it's zero.

When n -> infinite, p -> 0, but it never reaches 0

If p=0, that means there has to be some maximum game length, which is smaller than infinite. What might that be?
If the maximum game length is not infinite rounds, what is it then? Say any number, there is allways 1/36 probability that it goes one round further.

If we count all the possible ends when one player doesn't throw 5+5, we get an infinite amount of games.
The game is at point when endless double 5 results endless game. After that, for every double 5, there is at least 21 other possible games that are different from that particular double 5 game. Wasn't the purpose to count number of possible games? If it is possible to throw 5+5 infinite amount of rounds, there has to be at least 21 * infinite = infinite possible games.

Chessmaster1000: How do you conclude that.......?

Trivial. Suppose you don't get an infinite number of finished games. Then there should be a finite number. Take the one which took the most moves, say R moves. But then your finite set of games didn't include the game that finished after R + 1 rolls of 5-5 by both sides. So, the assumption that there are a finite number of games is false.

AbigailII: You got it wrong.
Correct! I know.....I felt that my example was wrong, but i never really believed it was. Seems stupid right? It was just a matter of not thinking about it a bit more........

I will add later something to the very interesting point of: "Say there are a finite number of games, call this number M. But that could not have included a game that reached the position I described above and then continued with M rolls of 1-1 on both sides, followed by a roll of 6-6. Ergo, there's no limit on the number of different games.

Wil: For every move to the infinitum, the probability is 1/36, why would it be smaller at some point?
For every move it's 1/36.
For 2 moves to happen is (1/36)^2
For 3 moves to happen is (1/36)^3
For an infinite number of times it's zero.

If we count all the possible ends when one player doesn't throw 5+5, we get an infinite amount of games.

How do you conclude that.......?!?!?!?

Chessmaster1000: that game would be one single game

If we count all the possible ends when one player doesn't throw 5+5, we get an infinite amount of games.

Chessmaster1000: You got it wrong. In the limit, the chance that you roll 5-5 "for ever" goes to zero, that's right. So, the chance for an infinite long game is zero. But that's not the same as an infinite number of games. Here is how it goes:

Take the following position: both players have 13 pieces off. White has its two remaining pieces on his 6 point. Black has its two remaining pieces on white's 5 point. White to roll.

If white rolls 6-6, the game is over. Call this game 1.
For game two, white rolls 1-1 (can't move). Black rolls 1-1 (can't move either). White rolls 6-6. End of game. This is game 2.
For game 3, the sequence goes: white rolls 1-1, black rolls 1-1, white rolls 1-1, black rolls 1-1, white rolls 6-6.
Or more general, for game N, both white and black start with N-1 rolls of 1-1 (this chance is not zero), and then white rolls 6-6.

Say there are a finite number of games, call this number M. But that could not have included a game that reached the position I described above and then continued with M rolls of 1-1 on both sides, followed by a roll of 6-6. Ergo, there's no limit on the number of different games.

Chessmaster1000:1st)The probability that both sides will roll a 55 an infinite number of times is exactly zero!

For every move to the infinitum, the probability is 1/36, why would it be smaller at some point?

It's question of possibility not probability.
Ie, if we start counting all the possible games:
....Oh, on this point player 1 can throw 1+1, 1+2 ... 5+5,.. hmm.. let's look more carefully this 5+5. Oh, on this point player 2 can throw 1+1, 1+2, .. 5+5.. hmm.. let's look more carefully this one.. Oh, on this point player 1 can throw ..

playBunny: Both sides roll 5-5 ad infinitum

1st)The probability that both sides will roll a 55 an infinite number of times is exactly zero!
With other words : The game would end in a finite time if every single move is made in finite time.....

2nd)Even if the game will continue with an infinite number of 55 (although this can never happen as i said), that game would be one single game and this doesn't help us in the question of how many Backgammon games exist? Finite or infinite? It's another different subject.......
Well it actually "connects" with the AbigaiIII's theorem, but as i believe this theorem is wrong you understand that.....

Chessmaster1000: But that really proves that the number of different Backgammon games are infinite.....?

I agree, and if you have ever tried anti-backgammon, there is no question about it.. ;-)

Hmm i understand. But you must have a mistake in your previous post. Please correct it......

You wrote:
The number of different backgammon games is finite if, and only if, there's at least one game with a position that repeats itself.

I guess you should replace finite with infinite. Right........?

And of cource an easy proof that these special positions exist, is to choose M=N+1 and have both opponents at the bar and choose such a dice roll that doesn't get any of the 2 from the bar, for 2 consecutive rolls........

But that really proves that the number of different Backgammon games are infinite.....? At a first glance it does, as k can go to infinity but perhaps this is not critical......
I will think about it and answer later......

Chessmaster1000: All pieces on the ace point except for two men each. Black has the 15th piece on the Bar and White has the 15th piece on the 5-point. Both sides roll 5-5 ad infinitum.

By "all possible, different Backgammon games" do you mean the variants? In which case we need a list so that we're talking about the same thing. From VogClub I know a few variants:
Tapa, Narde (Feuga) and Crazy Narde (Gul Bara) are finite as there is no sending back to the bar.
Longammon, Nackgammon and Acey-Deucey are the same as Backgammon.
Hypergammon is, of course, ripe with infinity.

Chessmaster1000: AbigailII: there's at least one game with a position that repeats itself.

Can you explain in more details this.........

Eh, you have a game, and the positions (that is, where the pieces are on the board, and whose move it is, and if you have a cube, the value of the cube) after move N and M (for N and M not equal to each other) are the same. If that can happen in a game, you have an infinite amount of different backgammon games (if the position after move N can occur after move M again, it can also occur after move 2M - N, 3M - 2N, 4M - 3N, ..., kM - (k-1)N, k >= 0).

If positions cannot be repeated, the number of different games is finite - as the number of different positions is finite.

WhiteTower:
It depends on how you define "similar"........


AbigailII:
there's at least one game with a position that repeats itself.

Can you explain in more details this.........

Chessmaster1000: The number of different backgammon games is finite if, and only if, there's at least one game with a position that repeats itself. And that's not hard to construct - just take a game with both sides having 14 pieces on their 1 spot, and the remaining pieces not having broken contact. Then repeatedly, knock of the single piece. Eventually, the position must repeat itself.

Chessmaster1000: Yes, calculating the upper bound of possible games in GC does sound similar to what you are asking, doesn't it? :)

WhiteTower: About infinite or not BG games, ask Grim Reaper, he had some fun calculating similar cases for Gothic Chess :)

Similar cases? No! Nothing similar as i remember......
It was just a try to calculate the upper bound of possible arrangements if no pieces were captured. The upper bound and not even the absolute number.
And of cource it can be easily shown without any calculations that the number of possible Gothic Chess games is finite. For the moment i can't easily show that the same exists for Backgammon.......

Chessmaster1000: About infinite or not BG games, ask Grim Reaper, he had some fun calculating similar cases for Gothic Chess :)

As for Fencer's Law - it is as you say, and ends up being in the same way that Anubis existed both as an Ancient and as a Goa'uld (for Stargate fans!)

playBunny: Rule 19 in Connect-4 8x8? lol. What's that one about?

It's a set of rules and procedures that i use, trying to solve with a "computerized" way, the Connect-4 8x8 game............

And about the Fencer's Law. It's obvious that Fencer wanted the Backgammon game to be played correctly, but a programming bug created all these conversations........It was not his intention to play the game with other rules......
Also the Backgammon rules page here at Brainking is wrong.......When do you want to define the rules of a game you should include all possible situations possible.......

By the way, a question that came to my mind now and in the first 2-3 minutes i tried to think about it, i've been confused.
Do you know if all possible, different Backgammon games are infinite or not?
I will think about it at night but if you know the answer and help me save some time from trying to find it, i would be very thankful.....

Wil: DG is much smarter and does take care of all such cases, to the point of pre-playing moves to speed things up. But DG is only for BG, so it can afford the extra system load, whereas BK... ;)

WhiteTower: As I pointed out, this question is purely philosophical.. Originally Chessmaster asked, which move would be correct according to backgammon rules and my opininon is that using only one is not honoring the rules.

But your are all right, it doesn't affect the result, so I'll stop whining about this. You all can peacefully keep breaking the rule at that situation.. ;-)

Walter Montego: I could do just the same than if you break the same rule elsewhere in the game. I could ask you to make your move again.. ;-) Well, I guess I really wouldn't mind...

Hmm.. GNUbg offers only one choise on that case.. I wonder what choises for example Dailygammon gives..

Walter Montego: Exactly - turning to Wil> - either accept the status quo here and shut up about it, or don't accept it, shout at the top of your (virtual) voice and watch it get lost in the wind ;) (unless your stars are lucky and you make more sense than everyone else here and your request gets some results)

Wil: Yeah, right. Suppose you and I were playing a game and this very example happened in it and I played the four to win the game. If you object to it, just what can you do about it? That's what I thought, now pay up and let's get another game going.

Chessmaster1000: I agree. By "Time we had some" I meant the world. ;-)
Rule 19 in Connect-4 8x8? lol. What's that one about?

WhiteTower: Reality? Where you bin? You've missed a whole huge debate about Fencer's law. ;-p

And back to reality now: Here there is only Law 0: Fencer's Law :)

playBunny: Personally i don't care much about Law 20, but for my rule 19 that is getting on my nerves and prevents me to go into the next step at solving Connect-4 8x8.....

But even if we clarify what Law has the priority it doesn't matter much, as it will be written by a book and not an official, international Backgammon organization.........

Chessmaster1000: It's about time we had some then. We desperately need to know whether Law 20 is the most important in spite of Law 13 overiding Law 17.

Chessmaster1000: ...and that's because there is the element of chance in it. Can anyone tell us if there is an International Roulette/Slot-Machine/etc. Federation? :)

grenv: Pedantic? No, it's just a logical fact......
And for all: you should stop looking for an international set of Backgammon rules, as there isn't such thing........FIDE is for Chess but for Backgammon there isn't any.......

AbigailII: Okay. Let me answer your points as directly as I can:

[For the last time. But you don't seem to be able to grasp the concept that rules should be complete, and not refering to things that aren't defined.]

I fully accept that the rules should be complete. I fully accept that the rules should be correctly implemented. There should be no ambiguity and there should be no bugs. Similarly people should always be fair and the sun should always shine when we want it to. In other words it doesn't matter what should be, it matters what is. And I am only arguing about what is - the bug and the (English) rules as written. The grey area is in what these things mean and that's what this debate is about.

So:

1) [The rules of backgammon (as stated on THIS site, not rules of backgammon defined by some other identity)]

I am talking about the rules on THIS site and no other.

2) [.. nowhere state that if it is possible to move with both die, you have to do so. (This is your MDU rule).]

There IS an MDU rule - it's the one that the Fencer-acknowledged bug fails to enforce.

3) [Considering that you have to pass if there is no legal move available (no my words - read the rules), and you call this situation "impossible moves", it seems that what you call "impossible moves" is what the rules call "no legal moves".]

The rules say that the player "must pass" if they "cannot make a legal move". The use of the word "legal" in that sentence is irrelevant. It could simply say "cannot make a move" and it would still be correct. There would be no change to the meaning, either literally or by implication.

Here's a choice for you:

3a) There is a distinction between possible and legal. It's impossible to move past a prime. Legality doesn't come into it. It's impossible to come off the bar into a closed table. It's true that there are no legal moves but that's because there are no possible moves. The impossibility of a move precludes legality; you cannot judge the legality of a non-move.

Or let's say that you can't bring yourself to agree with 3a).

3b) All impossible moves are in the illegal moves category. It should be obvious that there are possible moves which are also illegal. Illegality would then be a concept that applies to all impossible moves and some possible moves. These last constitute two separate sets.

4) [Again, the rules do not define any MDU rule.]

Correct, the rules do not explicitly define an MDU rule. We have agreed on this several times now. ;-p

5) [Therefore, the "no swapping possible if there's no legal move for the second die" isn't referring to the MDU rule, because there is NO MDU rule.]

It's quite the opposite, I would suggest. This no 'Swap dice' link shown when there's no legal move with the second die" covers two situations which I've shown are separate (see 3a) or 3b), whichever you prefer).

5a) The first is when swapping the dice makes no sense because using that dice value is impossible.

5b) The second is when swapping the dice would lead to a possible but illegal move.

6) The obvious question is "what are these classes of possible but illegal moves?". The non-MDU-compliant moves are the only known class so far. And I invite you, yet again, to come up with another because my argument will collapse if you do. Go for it! ;-)

7) The sentence must therefore imply the MDU rule.

Conclusion:

The MDU rule is a BrainKing rule that is both implied by the written rules and acknowledged as a behaviour that the backgammon server should enforce but doesn't. Non-MDU-compliant moves are therefore against the rules, illegal, not to be done, yada, yada, yada.


Let's try a different tack.

Q: Does the MDU bug exist?
A: Without a doubt. It's clearly documented in the bug tracker (and, besides, it's what triggered these debates!).

Q: Is the bug really about the MDU rule?
A: Absolutely. Both of the original instances in the bug tracker as well as Wil's new example are cases of MDU rule.

Q: If this bug is a failure to enforce the MDU rule, doesn't that mean that the MDU rule is part of the BrainKing rules?
A: As Spock would say: "Logic dictates this".

Q: But if the rule isn't explicitly stated on the rules page, doesn't that mean the rule doesn't exist?
A: No. As the previous question indicates, the existence of the bug implies the existence of the rule. It should be assumed that the written rules are lagging behind and need updating.

Q: Ah, but if the rule isn't written and there's a bug which means the rule isn't actually enforced, then surely there is NO MDU rule.
A: You can certainly argue that point but it doesn't negate the fact that the bug and the written rules imply the MDU rule.

Q: Implied? Only implied?? That's not strong enough for me!
A: Ain't nuthin' I can do 'bout that!

LOLOL.

Wil: That is the most pedantic point I've ever heard. In that example you win the game either way, so a clarification of the rules in that situation is both irrelevant and silly.

I can't think of a bearing off example where it actually matters. Even with 2 left on 2 and 3 with an opponent blocking 1, and you roll 6-1. In this case you need to move 3-2, 2-off. But even if you move 3-off you are left in the same position, so who cares?

Walter Montego: This to me means either way of moving the last man off is OK by the rules.

Not for me.. Law 17 definitely says "Law 13 applies here as in all other situations". Doesn't that mean that you cannot bear off by breaking the Law 13, "Plays must be made for both dice if possible"?

>If I use the 4 and bear my last man off, the game
>is over by Law

As I pointed out, according to Law 17 referring to Law 13, you cannot bear out with 4 without first using 1, because you CAN use both dice.

playBunny: Law 13 says:

Plays must be made for both dice if possible. Either number may be played first. If either may be played, but not both, then the higher number thrown must be played.

Law 17 states:

When in a position to bear off, you may bear off a man from a point corresponding to the number on a die thrown, or from the highest occupied point which is lower to the number indicated by a die. If a number is thrown for an unoccupied point, no man below can be borne off, using such number, while any man remains on a higher point. You are not required to bear off a man if you are able to move a man forward on the board. Law 13 applies here as in all other situations.

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXx
The authors pause here to show what ChessMaster1000 was talking about when not necessarily using all of the pips. What he said is correct according to these rules. You do not have to use the pips to maximum gain as long as you use both dice in either order. They show an example of when you would want to play it one way as compared to the other.

OK, this may or may not answer the question about how one moves his last man off the board. You guys will have to appeal to the Supreme Court while I will refer you to Law 20 in the scoring section:

A game is won by the player who first bears off all of his men.

This to me means either way of moving the last man off is OK by the rules. If I use the 4 and bear my last man off, the game is over by Law and I am no longer able to make any more moves. If I use the 1 to hit my opponent's blot and then the 4 to bear my last man off, I have again won the game and it is over. It is clear to me that both are permissable without any of this illegal move stuff I keep reading about. I could see an argument if one made an illegal move that won him the game that was against the rules, but that isn't the case here. In NFL and NCAA Football, even with the clock at zero time left, the game will continue if the defense had a penalty charged to it and the offensive side accepts the penalty. I imagine there's other sports and games that have similiar situations too.

The game's over. Now pay up, and let's get another game started. :)