Tips cantstop

The numbers near the centre 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! In turn, 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 4 moves on 6/7/8.

The longer you can delay putting the third black token on the table, the longer you can 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.

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 that 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 2 players (especially 3), try not to fight the same opponent on every track you go for.

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

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 2 if all the black tokens are on odd columns, and subtract 2 if they are on even columns. Add 4 points if all the tokens are on value less than 8, or if all three are on values greater than 6.

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 / 12	13.2%
3 / 11	23.3%
4 / 10	35.6%
5 / 9	44.8%
6 / 8	56.1%
7	64.3%

Combinations of temporary markers

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

For some combinations of 3 chosen temporary markers, the following table shows (as long as none of the columns is 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, i.e. breakeven progress * probability to fail = expected progress per roll. I.e. at breakeven progress, you 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, 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 got one marker in 2, 7, 12 each, this method says it's best to stop right away, as your progress is already 0.81 (above the breakeven progress)