What is a Priming Game?
In a priming game, you make several consecutive points to block your opponent’s rear checkers from escaping. The starting position already gives you the eight and six points. You seek the four and five points because they are so useful in many kinds of games. Thus you may often build a priming game around your four through eight points. Your most important points for blocking the escape of your opponent’s back men are those that are five or six pips away.
A priming game is most effective when your opponent has two or more trapped checkers. When he has only one man back, it is often better to attack that lone blot to prevent escape.
Prime-versus-prime positions (in which both sides have primes and trapped checkers) are highly volatile, with gammon chances for both sides. Delicate timing problems and difficult plays abound for both sides. Cube ownership confers a considerable advantage.
||Black doubles to 4. Should white take?|
To bring the win home, black must roll a 6 to escape his back man. Unless he does so in time, his board will crack. Then white may enter and attack black’s back man to win quickly with a powerful redouble.
How much time does black have? He can play 23 pips with the checkers on the 8 and 17 points. On the average, he will have four rolls in which to roll a 6 before he must break an inside point. (Though the average roll is 8.17 pips, the average roll not containing a 6 is only 7.20 pips.)
We may calculate the chances of rolling a particular number (in this case, a 6) in n shakes by subtracting the nth power of 25/36 from 1. Thus the chances of rolling a 6 are:
- In 1 roll: 1 −
- In 2 rolls: 1 −
25 × 25 36 × 36
- In 3 rolls: 1 −
25 × 25 × 25 36 × 36 × 36
- In 4 rolls: 1 −
25 × 25 × 25 × 25 36 × 36 × 36 × 36
- In 5 rolls: 1 −
25 × 25 × 25 × 25 × 25 36 × 36 × 36 × 36 × 36
Does 77% therefore represent black’s winning probability in Example 1? If so, that would mean white had a bare take. But black has good winning chances even if he doesn’t escape before cracking.
Suppose, for example, black cracks on the fourth shake. He’ll break his six point before any other, of course. But white is a 25-to-11 underdog to roll an immediate 6. Black still gets a chance to escape with a 6.
Failing this, black won’t necessarily lose. Because his 5’s have been killed, the crashing of his board decelerates. White may continue dancing on black’s five-point board until (finally!) black escapes with a 6.
I’d say black wins about 85% of the time. Furthermore, white’s three outfield checkers, plus his open one and two points, put him in some gammon danger. Perhaps black’s 85% wins break down into 79% plain games and 6% gammons.
White should pass.
||Should black double to 4? Should White take?|
Though black’s decision to redouble or keep the cube precedes white’s decision to pass or take, the logical order for evaluating cube decision is pass-or-take first, turn-or-keep the cube second. Here black has two back men to free, substantially more difficult task than freeing one back man in Example 1, so it is clear that white should take.
Pass-or-take decisions can be resolved by estimating the taker’s winning chances, but cube-or-wait decisions require comparison of equities both ways.
- Suppose black redoubles to 4. With the checker on the 17 point, he can play 16 pips, giving hime an average of three shakes to roll an escaping 6.
As shown above, black’s chance of success is 67%. That will bring him to the position in Example 1, perhaps better, perhaps worse, depending on how soon he rolls the 6. But if black fails and starts to crack while he has two men back, white can redouble and win with the cube. The redouble thus yields an equity of0.67 × 0.67 × 4 + 0.67 × 0.06 × 8 − 0.67 × 0.15 × 4 − 0.33 × 4= 2.1172 + 0.3216 − 0.4020 − 1.3200 = +0.7168,or about +0.72.
- Suppose black keeps the cube at 2. Then he will reach the position in Example 1 (more or less) 67% of the time and win outright by redoubling. The other 33% of the time, his cube ownership will let him finish the game.
Rollouts show that even after starting to crash, black will win about 20% if these games. Thus his equity is0.67 × 2 + 0.33 × 0.20 × 2 − 0.33 × 0.80 × 2= 1.340 + 0.132 = +0.944,or about +0.94.
By a wide margin (about 2/9 of a point), black should keep the cube.
His best strategy is to wait until he rolls a 6 (without breaking his board) before redoubling, winning all the time when he does and some of the time when he doesn’t. A premature redouble costs black some of the games in which he frees one back man before crashing as well as some of the games in which he crashes.