A hit checker goes to the bar, and until it comes back you have no move at all. Not a bad move — none: the other fourteen checkers stand still whatever the dice show.
It comes back into the opponent’s home board. The die says onto which point: a one leads to the twenty-fourth, a six to the nineteenth. A point where the opponent holds two checkers or more is closed, and that die is no good.
The whole reckoning follows. You cannot come in only when both dice land on closed points.
How many rolls bring the checker back
| Points closed | Rolls that enter, of 36 | Chance | Turns to wait |
|---|---|---|---|
| 0 | 36 | 100 % | 1 |
| 1 | 35 | 97 % | 1.03 |
| 2 | 32 | 89 % | 1.12 |
| 3 | 27 | 75 % | 1.33 |
| 4 | 20 | 56 % | 1.8 |
| 5 | 11 | 31 % | 3.27 |
| 6 | 0 | — | — |
Three closed points look menacing and cost little: three turns out of four the checker comes straight back.
Every next point costs more than the last
Here is where the figures stop being dull. Take the difference between the rows:
1, 3, 5, 7, 9, 11.
The first closed point takes one roll out of thirty-six. The sixth takes eleven. Eleven against one: the same point, the same checker, an elevenfold difference.
So “close one more” early and late are two different acts. The first two points barely interfere: with two closed the checker comes in nine times out of ten. The fourth starts to work, the fifth decides, and the sixth ends the game.
White — on the bar: 1
Four closed points. The three and the two both lead onto closed ground, the turn is lost whole — and so it goes in sixteen rolls out of thirty-six.
Two on the bar is not twice as bad but four times
A second hit checker shifts the reckoning harder than it looks. For both to come back, both dice must land on open ground:
| Points closed | Both enter, of 36 |
|---|---|
| 0 | 36 |
| 1 | 25 |
| 2 | 16 |
| 3 | 9 |
| 4 | 4 |
| 5 | 1 |
With three points closed one checker enters three times out of four, and both enter one time in four. With four closed, one enters more often than not, and both one time in nine.
Hence a plain rule at the board: once the opponent holds three closed points or more, a second hit on you costs roughly twice what the first one did.
A closed board is not forever
Six closed points mean no roll brings you in. But the opponent can hold them only until he starts bearing off: a checker taken off opens a point, and the point is yours again.
So turns lost against a closed board are not a defeat but a count: the opponent gathers points while you stand, and the only question is how many he manages before the first point opens.
What this count does not measure
Coming in is not the same as getting away. The checker returns into the opponent’s home board, the farthest corner there is, with the whole road ahead of it. It may land under a fresh shot. It may instead hit a blot itself and turn the game around. The figure in the table says only whether it moves at all.
While you wait, the opponent does not stand still. The table treats the closed points as fixed, and at the board their number grows: every turn you lose is a free turn for him. Thirty-one per cent against five closed points is true for this roll, not the next one.
A chance is not a price. Twenty rolls out of thirty-six against four closed points look bearable, but a turn lost in the thick of the game and a turn lost in a race cost differently. The table gives frequency, not price.
What to do with it
When you are the one closing. Do not count the points — count which one it is. The third and fourth give more than the first two together, and they are where the turn is worth spending when the choice is between closing and advancing.
When you are the one on the bar. Up to three closed points play as usual: the checker comes back three times out of four. From four on, start counting not your chances of entering but his chances of getting on — how many turns he wins while you stand.
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