Say you have 2 checkers on the 6 point, 2 checkers on the 5 point, 2 checkers on the 3 point, 2 checkers on the 2 point and zero checkers on the 1 point. You roll a 5 - 1. What is the best strategy for moving the checkers?

I think I read once that the best thing to do in this case is to use the 1 to move from the 6 point to the 5 point.

Thanks for any input.

Well, the general principle in the bearoff (without contact) is:

If you are a favourite to win the game, you should work on a better distribution. This will improve your chances bearing off 2 checkers with most rolls. In your position the move 5/off 2/1 is improving the distribution as it only leaves a gap on the 4-point.

If however you are the underdog, you should search for moves, that let you bear off more checkers if you roll some jokers (= doubles). You have to optimize for the miracle, that might happen. For the given position, 6/off is a bit better, letting you bear off 3 checkers with a 55 (instead of only 2 if you move 5/0ff 2/1).

So the answer to your question depends on the opponents position.

But, take it easy. In this case, the difference between the two moves is marginal.


Hardy    

hmmm Hardy gives a better answer  

If you think that you are better than your opponent you should play 5-off 2/1, if you are the underdog you should play 6/off and if you think that you are both equal it depends on your expectations. Actually, if you are risk lover you must choose 6/off and if you are risk averse you must choose 5-off 2/1 (In the case that you are both equal)

Well, in more than 99% of all cases it IS correct to bear off a checker if you can do so. But there are some positions, where bearing off is an error and you better smooth the position. Have a look at this position (DMP, so no cube):



Here it is much better to use the 2 to move 4/2. Otherwise you have a gap you will never be able to fill again. And any 2 will not allow you to bear off a checker.

Winning chances after the two possible moves are:

6/off 4/2: 71,20 %
6/off 2/off: 70,09%

So by playing 2/off you give away more than 1% of match winning chances! A huge blunder.


Hardy  

 

Hardy

I've been thinking about this one. You've got me mystified

Give me a series of rolls where you'd be off the board before me by going 4/2 rather than 2/off.

How is the 22 worse?  If you move the 6 off, 2 off, then roll 22, the one from the 4 goes off - and 2 from 3 to 1, which is EXACTLY the same result as if you had moved 6 off, 4-2 in the previous move?

Perhaps I should add two remarks, I didn't state clearly enough above:

1. Counting the resulting moves required to bear off is an important thing! In the discussion above its either 9 or 10, which makes no difference in terms of rolls required to bear off. And with 4/2 you are likely to miss no single roll in the bear off process (only if you roll more than two times a 2).

2. The generals principle as stated in the very beginning of this discussion is still valid: If you are an underdog to win, you are more likely to be correct to bear off as many as you can. Lets modify the position discussed a bit on the opponents side, so he is a favourite now to win:



Now your match winning chances are:

6/off 2/off: 27,84%
6/off 4/2: 27,64%

Now it's slightly better to bear off.

Yes, I know, backgammon is confusing sometimes. But I can't help it  


Hardy  



 

snowjake, you are a hopeless case. I have drawn a board where NOT bearing off is better than bearing off. I did explain why and I gave a precise descritpion of rolls, where you are better of having played 4/2 instead of 2/off.

In the given example you probably require 5 rolls to bear off. If just two of those rolls contain a 2 (16% probability), you require 6 rolls instead if you played 2/off instead of 4/2. When playing 4/2, you still require 5 rolls to bear off, even if two of your 5 rolls contains a 2. Just count the rolls required.

In the give example you don't loose anything by playing 4/2. You gain instead. Get your board, try it, count the moves.

And again: Get the recommended backgammon software. And use it.


Hardy  



 

Sorry snowjake, I wont be able to satisfy your request. You want me to guide you to every individual roll until you are convinced that playing 4/2 is better than bearing off??? That's impossible!

With 9 or 10 black checkers left, the match will be over in about 10 moves (5 moves of WHITE, 5 moves of BLACK), or a bit less if one of the player rolls some doubles. For any roll there are 21 possible different outcomes of the dice (11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 33. 34. 35, 36, 44, 45, 46, 55, 56, 66). If we ignore, that some moves can be played differently, that are about 21 ^ 10 different moves to go through (=16.679.880.978.201 moves). If the game is over after 8 rolls it's still 21 ^ 8 (=37.822.859.361 moves). That's a bit too much - even for me.

How do you work it out? In the early days of backgammon (70s and 80s) you conducted manual rollouts. I.e. playing this position to conclusion for many hundred times. Each time you wrote down the result. At the end you had a rough estimate of the percentages. A very boring procedure. Since the middle of the 90s there is modern, reliable backgammon software available which can do the rollouts for you. You ask the software to conduct the rollouts, the software will tell you which move results in which percentage of wins and losses. With the help of that software you can identfy which move is the best. Thats what I did. And I added the general principles that apply for the bearoff. Either you believe me (and backgammon software), or stay with your preferences.

I wish you much joy in working through that position yourself.


Hardy  
 

Snowjake, this is not the way it works. Sorry.

It doesn't help to show one particular sequence of rolls where one move is better than the other. That doesn't prove anything. You have to compare it to all other possible combinations.

What I can tell you is:

If during the bearoff you roll two times a 2, you will finish your bearoff one whole move earlier when you had played 4/2 than if you had played 2/off. If that helped you to win, depends of the rolls of your opponent and the rest of your rolls. Can you follow me in that reasoning?

If I can't persuade you, my patience inventory is exhausted. Try to google for similar discussions (keywords: "backgammon bearoff smooth"). Perhaps others can persuade you.

Hardy