Tips cantstop

The numbers near the middle of the board are easier to roll. If you manage to get your tokens on 6-7-8 or 5-7-8 or 6-7-9, it's highly likely that you'll be able to advance. Time to get greedy and make the most of it!

If you have advanced on 2, 3, 11, or 12, and have all three black tokens on the board it is often advisable to stop immediately. Remember, a move ahead on 2-12 is worth about four moves on 6-7-8.

The longer you can delay putting the third black token on the table, the longer you can (usually) safely continue rolling. For this reason, on your first few rolls it is usually preferable to advance on one track only (especially if you can do it twice) than on two different tracks.

Play to manage your risks. If you're far ahead, play it a bit safer. If you're far behind, the best you can get by avoiding risks is a respectable defeat.

Remember you can't use tracks won by any player. This includes the ones you've won, and the ones you're about to win if you stop. Thus in the later stages of the game, it's easy to get stuck and lose a turn.

Pick your battles. In games with more than two players (especially games of three players), try not to fight the same opponent on every track you go for.

Never ever assume you'll win a track, even if you only need to move one more, especially near the middle of the board.

The Rule of 28

A researcher found an heuristic to help playing the game (Michael Keller, 1986). Each column has a number of points:

• 2 and 12 are worth 6 points
• 3 and 11 are worth 5
• 4 and 10 are worth 4
• 5 and 9 are worth 3
• 6 and 8 are worth 2
• 7 is worth 1

When you progress a black token on a column, add the number of points to your total. When you put a black token on the table for the first time, double the value. Whenever your total is equal or more than 28, stop.

For further refinement, add two if all the black tokens are on odd columns, and subtract two if they are on even columns. Add four points if all the tokens are on value less than eight, or if all three are on values greater than six.

Remember this strategy is not flawless.

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Probabilities

Probability of Getting a Particular Number

The following table shows the probability to get a particular number on the following roll:

Number	Probability
2 and 12	13.2%
3 and 11	23.3%
4 and 10	35.6%
5 and 9	44.8%
6 and 8	56.1%
7	64.3%

Combinations of Temporary Markers

On the first roll it is fair to try to maximize expected progress with progress as the "fractional" number of closed columns, i.e. (number of 2s and 12s) / 3 + (number of 3s and 11s) / 5 + (number of 4s and 10s) / 7 + (number of 5s and 9s) / 9 + (number of 6s and 8s) / 11 + (number of 7s) / 13

For some combinations of three chosen temporary markers, the following table shows (as long as none of the columns are won):

• The probability to advance on each roll (which is 100% - the probability being unable to continue)
• The expected progress on each roll (as defined above)
• The resulting breakeven progress
• The probability to fail equals the expected progress per roll (i.e. at breakeven progress, you can expect to gain as much as to lose by continuing to roll. So, if your progress is more, stop. And, if your progress is less, continue to roll.

Chosen Columns	Probability to continue	Expected progress	Breakeven progress
[6, 7, 8]	92.0%	0.115	1.44
[5, 7, 8]	91.4%	0.119	1.39
[5, 6, 7]	88.7%	0.113	0.999
[5, 7, 9]	85.3%	0.116	0.788
[5, 6, 8]	89.5%	0.12	1.14
[4, 7, 8]	90.3%	0.123	1.27
[4, 6, 7]	88.6%	0.12	1.05
[3, 7, 8]	89.3%	0.125	1.17
[2, 7, 8]	89.0%	0.13	1.19
[2, 6, 7]	86.4%	0.126	0.926
[3, 7, 11]	77.6%	0.13	0.581
[2, 7, 12]	78.1%	0.137	0.625

Some conclusions:

• With 6, 7, and 8 you should/can roll the furthest. On average, it's best to roll 13 times with this combination, depending on what you get.
• 5-7-8 or 6-7-9 are actually very similar.
• 5-6-7 or 7-8-9 are a lot worse than that (even though these lines are as long as the previous ones)
• If you have markers on 2, 7, and 12, this method says it's best to stop right away as your progress is already 0.81 (above the breakeven progress).