The alert reader by now has determined the first article in this series is neither ground breaking nor original. It is simply an expansion of the work of Douglas Zare in his article "The Half Crossover Pipcount".
What makes it the Northern Michigan Pipcount is how it is used. Like most players, I have a medical degree and at least 3 semesters of college calculus. Like most players at 3 a.m. and after three beers I have trouble subtracting the year of my birth from the current year and coming up with my age. I needed to find a way to get a pip count easily and answer some basic questions, like "Am I ahead enough to double?" or "Can I take this double or should I pass?" or most importantly "Should I finish this game or get rid of my last beer?"
To recap the first article, to get a pip count, you divide the board into half crossovers with 1/2/3 the first (each checker counts as −1), 4/5/6 as home (each checker counts as 0), then 7/8/9 (1 each), 10/11/12 (2 each), so on and so forth to 22/23/24 (each has 6 half crossovers to get home). The bar is 7 half crossovers from home.
A backgammon board with half crossovers marked.
We determined the formula to get a rough pip count was (# half crossovers × 3) + 75, so that in the opening position there were 30 half crossovers for a total of (30 × 3) + 75 = 165. We then learned to adjust the rough count by looking at the midpoint of each half crossover and adjusting for checkers 1 pip below or 1 pip above the midpoint to get our final pip count. Although this is actually workable over the board, is there an easier way to use it? Sure enough, there is.
Backgammon players like reference positions and formulas, so here are a couple of really useful ones:
- 25 half crossovers = 150 pips, 15 = 120 pips and 5 = 90 pips
- −5 half crossovers = 60 pips, −15 = 30 pips and 0 = 75 pips
- 8% of your pip count is about (half crossovers/4) + 6, 12% is this number × 1.5
- Each half crossover different from the reference positions changes the pip count by 3
- Finding the difference between your half crossovers and those of your opponent allows you to calculate roughly how far you lead or trail in pips (just multiply by 3).
Thus if I have 17 half crossovers, I know this is 15 half crossovers plus 2 half crossovers. Since 15 are 120, and my 2 extras give me another 6, my rough pip count is 126. By the same method, 23 half crossovers is 144 and 32 would be 171. If you have 22 half crossovers and your opponent has 29, you are about 21 pips ahead in the race (and in a straight race your opponent has a pass with anything over 16 or 17). Again, very little math involved. Remember your midpoint corrections though and to take off 5 for every man taken off the board (the formula is based on 15 including the bar).
Now we can go back to a basic question. In a no contact position, if I do a count of half crossovers and find I have 12 and my opponent has 15, do I have a double? 8% of 12 half crossovers is about (12/4) + 6 or about 9 pips, and 12% is just shy of 14. 15 half crossovers for my opponent and 12 for me means I lead by 3 half crossovers, or 9 pips. Barely a double as I need 9 to turn the cube and probably just below 14 to cash.
Red has 12 half crossovers, White has 15 half crossovers.
But that is a rough count, how about the exact count? Recall from the Zare formula we needed to do a final correction to get an exact count. Say I find that I have to subtract 2 from my rough total. Well that adds 2 to my lead to make it 11. Now the final step, look at my opponentís groupings and adjust his final pip count. Say I have to add 3 to his rough count. That makes my lead 9 + my 2 + his 3 = 14. The actual pip count is 109 to 123 by the way.
By zeroing in on the actual pip count we quickly went from "I have a rough lead in the race" to "I have nearly maximally efficient double just a shade past your take point (in this case 12.7%)". Using the formal half crossover pip count we know we had a cash at 109 to 123. We were pretty sure we had a cash calculating our lead out at 14 with quite a bit less math. If our opponent was not as precise, using the full crossover count alone, he may have found he trailed by two crossovers, which is generally considered a take.
While the half crossover method and the Zare formula are designed to do pip counts, can we use the concept to help us with other problems? One such question is the gammon countthe number of pips we need to roll, at a minimum, to get off a gammon. When trying to save a gammon, the goal is to move all your runners to your 6 point, then take off one checker before your opponent bears off his last checker.
We can count the half crossovers needed to bring our runners home, then get the number of pips by multiplying that number by 3. Then correct our count by doing the midpoint adjustment for our runners only. Nothing new here, but we do need to make one final correction. Recall that by doing the above operations we find the pip count to bring all our runners to our 5 point (the midpoint of the home half crossover) but we only need to go to the 6 point. We correct this by subtracting 1 pip for each runner. Then we simply add the minimum number of pips it would take to bring one checker off.
Nine half crossovers gives a rough count of 27. Reduce this by 2 for men on 13 and 10. Subtract 3 more pips, one for each runner. Add 2 to take a man off the 2 point. Final gammon count is 24.
To summarize, we did our homework and learned the basis of the half crossover pip count. We derived the equation and then used it to formally calculate our rough and exact pip counts. Then we reviewed shortcuts comparing our actual half crossover count to a reference and did a little simple addition or subtraction to do the same thing. We compared our count to our opponentís count and calculated a lead. Then we refined that rough lead in series of simple adjustments (the midpoint adjustments) to get an exact lead. Then we looked at ways of determining 8 and 12% of the leaderís pip count. Then we half crossovers to calculate a gammon count.
There are many equations to learn in backgammon: Winning with the Doubling Cube by Peter Bell must have 2 dozen to help you evaluate your doubles. These include timing counts, gammon counts, Thorp counts, racing leads etc. all of which are derived in one way or another from the basic pip count. Beginning and social players know how to set up the board and roll the dice. Intermediate players know how to move their checkers efficiently. Advanced players understand the cube. The basis to understanding the cube starts with the pip count. The key to the pip count is a minimally taxing way to calculate it over the board without detracting from important considerations. Happy journey.