Backgammon Cube Handling

Cube Handling

Endgame close out: Spleischft formula

From:   Simon Larsen
Address:   larsen@ostenfeld.dk
Date:   8 September 1999
Subject:   endgame
Forum:   rec.games.backgammon
Google:   37d55210.0@news.k-net.dk

What is this article about? How to calculate you opponents chances of winning the game in situations like this: You are about to bear off, and you opponent has ONE checker on the bar and maybe he has borne some of his checkers off too. The End-game-spleischft formula: conditions: 1) Your opponent is on the bar (only one checker) 2) You have a closed inner table with spare checkers on 4,5 and 6. 3) your opponents innertable-checkers are smoothly distributed. 4) Its your turn. 5) Claus Thomsen is a killer on FIBS. If your opponent has borne x checkers off, his chance of winning the game is: ************************************************ if x is between 0 and 7: opponent wins 2*x*(x+1)/3+5 out of 100 games. if x is between 8 and 14: opponent wins 6*(x+1) out of 100 games. ************************************************ examples: +24-23-22-21-20-19-+---+18-17-16-15-14-13-+ | O O O O | | | | O O O O | | | | O O | | | | | | | | | | | | | | | | | | | | | O | | | X X X | | | | X X X X X X | | | | X X X X X X | | | +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+ the conditions is ok so: how many checkers has white borne off? -> 4 4*5=20 20*2=40 40/3=13 13+5=18 opponent wins 18% of all games Example 2 +24-23-22-21-20-19-+---+18-17-16-15-14-13-+ | O O | | | | O O | | | | O O | | | | O | | | | O | | | | | | | | | | | | | O | | | X X X | | | | X X X X X X | | | | X X X X X X | | | +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+ White has borne 6 pieces off. 6*7=42 42*2=84 84/3=28 28+5=33 BUT whites checkers is not smoothly distributed! the two checkers on 24 should have been on 22 - but since white dont need any extra rolls because of this we will not give him extra %'s. opponent wins: 33% The hard one :-) +24-23-22-21-20-19-+---+18-17-16-15-14-13-+ | O O O | | | | O O O | | | | O | | | | | | | | | | | | | | | | | | | | | O | | | X X X | | | | X X X X X X | | | | X X X X X X | | | +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+ white has borne 7 checkers off but white doesn't have a smooth distribution of checkers :-((( - We will have to create a equivalent-diagram. How? Lets say O would need one or two more rolls now than if his checkers was smoothly distributed. So I will add two checkers to his innertable and distribute his checkers. new equivalent diagram +24-23-22-21-20-19-+---+18-17-16-15-14-13-+ | O O O | | | | O O O | | | | O O O | | | | | | | | | | | | | | | | | | | | | O | | | X X X | | | | X X X X X X | | | | X X X X X X | | | +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+ now: white has 5 checkers borne off. 5*6=30 30*2=60 60/3=20 20+5=25 but the players checkers is placed worse than in the standard condition (spare checkers on 4-5-6) so we will give white 9% opponent wins: 25% + 9% = 34% In the last diagram I gave white 9% and here is why: this *NEW* table tells us the how many %'s we should give the opponent if our spare checkers is on 1-2-3 instead of 4-5-6: opponent has again borne (X) checkers off the BLUWDUCH table: (X), (5's to add) -------------------------- 1, 1% 2, 2% 3, 3% 4, 5% . 9, 9% (here is x=9 and we see that we need to add 9% :-D) . 14,5% interpolate between 4 and 9. ANd interpolate between 9 and 14 Final example to show the use of the formula for x between 9 and 14: +24-23-22-21-20-19-+---+18-17-16-15-14-13-+ | O O | | | | O O | | | | | | | | | | | | | | | | | | | | | | | | | O | | | X X X | | | | X X X X X X | | | | X X X X X X | | | +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+ white has borne 10 checkers off. x is now greater than 7, so we will use the other formula 6*(10+1)=6*11=66 (6*(X+1)) opponent wins: 66% What do you think? __ : this thing sucks. __ : power to the spleischft and bluwduch formula.

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