|
|
The Half-Crossover Pipcount The approximate pipcount is vital to correct backgammon play: "When ahead in the race, race. When behind in the race, don't race." Some times one needs more. The exact pipcount is extremely useful to guide one's doubling strategy in some situations. For example, in a race of more than 70 pips, if you are on roll a good rule of thumb is that if you lead by at least 8% then you probably have a correct double, and if your lead is at most 12% then your opponent probably has a correct take. It arises in many other situations as well. Since the pipcount is important, there have been a wide variety of methods developed to compute it over the board.
I'll outline a method I use for doing pipcounts which requires less mental arithmetic and mental rearranging of checkers than any other I have encountered. One doesn't even have to remember the numbers of the points! It mainly involves counting, though it also uses adding 2-digit numbers to 75 and multiplying 2-digit numbers by 3. As an added benefit, along the way one gets an approximate pipcount, which usually suffices. I call this the half-crossover method.
Step 1: Count the half-crossovers to bear in.
Counting crossovers is easy, but not refined enough for a good approximate pipcount. Instead, think of breaking each of the outer and inner boards in half, into triples: The first triple is 1-2-3, the next is 4-5-6, the third is 7-8-9, and the last is 22-23-24.
Let's assume that, rather than bearing off, your goal is to bear in. You want to bring your checkers to the 4-5-6 triple. Any checkers in the 4-5-6 triple get counted 0. The checkers in the 7-8-9 triple would have to move forward one half-crossover. Those in the 10-11-12 triple have to move forward 2, those in the 19-20-21 triple would have to move forward 5, and those in the 1-2-3 triple would have to move backward 1.
For example, from the starting position, you have 5 checkers already there, 3 checkers that need to move forward 1 (subtotal 3), 5 checkers that need to move forward 3 half-crossovers (3+15=18), and 2 checkers that need to move forward 6 half-crossovers (18+12=30).
If all you want to do is determine who is ahead, and by about how much, then you can stop here. If someone is ahead by 2 half-crossovers, then it is quite likely that they are actually ahead in the race. (This is quite different from counting crossovers. Being ahead by one crossover is a much weaker indicator of a racing lead.)
Step 2: Multiply this by 3, and add 75.
If all 15 checkers were on your 5, your pipcount would be 15*5=75. There are 3 extra pips for each half-crossover to bring a checker into the 4-5-6 triple.
From the starting position, you have 30 half-crossovers to go, so you add 3*30=90 to 75 to get 165 as your approximate pipcount.
Step 3: Subtract 1 for each checker on the front of its triple. Add 1 for each checker on the back of its triple.
What we did in the previous step was to assume that every checker was in the middle of its triple, that every checker in the 4-5-6 triple was actually on the 5. A checker on your 6 is one step further away, so add one for each checker which needs to be pushed forward to the middle of its triple. Similarly, a checker on your 4 is one step closer than we counted, so subtract one for each checker that needs to be pushed back to the center of its triple.
These points are the centers of triples, and count 0.
These are the backs of triples for red, and count +1.
These are the fronts of triples for red, and count -1.
In the starting position, you have 5 checkers on your 6 that need to be pushed forward minus 5 checkers on your 13 that need to be pushed back plus 2 checkers on your 24 that need to be pushed forward. 5-5+2=2, so one needs to adjust the approximation computed in Step 2 by adding 2: 165+2=167, the exact pipcount.
Feel free to cancel +1's and -1's that you can pair up. Since the 6 point and midpoint cancel, the modifications to the approximate pipcount are a bit smaller than one would expect by random chance. (Also, it is quite reasonable to count on your fingers. If you keep them under the table other people won't notice.) The net modification (between red and white) is usually less than 5 pips. |
|