Whilst we all struggle to understand the complexity of the middle game in backgammon, we continue to make very good progress in areas of the game where computers can assist us.. The obvious area for analysis is the non-contact endgame where the race becomes of the utmost importance. If we can produce and learn formulae to cover the majority of positions then those who can remember and apply those formulae will have an edge over those who have not studied this area of the game.
In this first article on backgammon races I am going to look at the easiest of the racing formulae that are available to us.
Until the 1970’s there really weren’t any formulae in common use and the vast majority of backgammon players would not have known a pip count if they’d run into one! With the surge of interest that took place in the 1970’s some order was finally imposed and the first rudimentary formula based on the pip count was developed.
Just in case you don’t know what a pip count is, it is the number of pips you must roll with the dice to bear off all your remaining men. The pip count in the starting position is 167 for each player.
The pip count is calculated by multiplying the number of checkers on each point by the value of the point and summing the total. In the diagram above black’s count is:
(2×24) + (5×13) + (3×8) + (5×6) = 48 + 65 + 24 + 30 = 167
To help understand the racing formulae a simple diagram is required:
- Leader’s Pip Count: The number of pips required by the leader to bear off all his checkers.
- Trailer’s Pip Count: The number of pips required by the trailer to bear off all his checkers.
- Doubling Point: The leader’s lead in the pip count is sufficient for him to be able to offer an initial double
- Redoubling Point: The leader’s lead in the pip count is sufficient for him to offer a redouble
- Point of Last Take: The point at which the difference in the pip counts is such that the trailer has a take but only just.Any increase in the difference would mean that the trailer must drop a double/redouble.
- Doubling Window: The pip count range within which double/take is the correct doubling decision.
The basic formula is very simple but for all that it is reasonably effective and certainly a huge step forward from the visual inspection techniques that preceded it:
Let the leader’s pip count be L.
Let the trailer’s pip count be T.
If T is 8% greater than L then the Leader should double.
If T is 9% greater than L then the Leader should redouble.
If T is 12% greater than L then the Trailer should pass the double or redouble.
This 8-9-12 formula, whilst very basic, still works well for long races and it is a good guide for the majority of races where the pip counts are greater than 50 (below that figure positional considerations normally, but not always, begin to exert a greater influence). For many years it was the only formula around and was used by all serious backgammon players.
Kit Woolsey restated a variant of this formula in his latest book “The Backgammon Encyclopaedia Volume 1”. Quoting directly: “The main measure for a race is the pip count. Other things such as smoothness, crossovers, and gaps also play a part, but the pip count is usually the first thing to be looked at. The general theory is that for medium to long races, the doubling window centres around a 10% lead. Two pips more advantage for the leader and the trailer has a borderline take/pass.Two pips less for the leader and the leader has a borderline double (for an initial double he generally needs one fewer pip). This formula is pretty accurate for most of the races until the final bear-off stages are reached, assuming neither player has checkers buried on the lower points.”
Let us take a look at a couple of examples:
In this first position black’s pip count is 100 (2×11 + 2×8 + 3×7 + 3×6 + 3×5 + 2×4) and red’s is 110 (1×11 +2×10 +3×9 +1×8 + 1×7 + 3×6 +3×5 + 1×4).
Black calculates 8% of 100 which is 8. Therefore he knows that if red’s pip count is greater than 108 (100+8) he should double and does so. Red calculates 12% of 100 which is 12. He can take if his pip count is less than 100 + 12 = 112 which it is, so he can confidently accept the double.
If we change this position slightly :
Here black’s position is the same but red is two pips better off so his pip count is now 108. Also notice that black owns the doubling cube.
Here is the math :
8% of 100 is 8 so black would have a borderline double if the cube was in the centre However, black is redoubling so needs to calculate 9% of 100 which is 9. As 100 + 9 is greater than 108 then black should not redouble but wait until he has improved his position.
In our two examples the percentages have been very easy to calculate. This is not always the case but after some practice the vast majority of backgammon players should be able to do the necessary arithmetic.
The 8-9-12 formula is relatively easy to apply and it will enable you to make the correct doubling decisions in literally thousands of race situations.
There are more sophisticated formulae that take positional factors into consideration and we will be taking a look at these in subsequent articles.