Tips cantstop - Revision history

Dubby wubby: /* The Rule of 28 */

2025-11-03T04:34:53Z

The Rule of 28

← Older revision		Revision as of 06:34, 3 November 2025
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	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.		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.		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. You can also consider targeting a certain player and challenging them for tracks to deny them points.

	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.		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.

JohnnyCyclops: Minor grammar

2023-07-13T18:07:22Z

Minor grammar

← Older revision		Revision as of 20:07, 13 July 2023
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	=== Combinations of Temporary Markers ===		=== Combinations of Temporary Markers ===

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

Mike Weiner at 05:12, 20 July 2022

2022-07-20T05:12:40Z

Mike Weiner at 05:09, 20 July 2022

2022-07-20T05:09:59Z

← Older revision		Revision as of 07:09, 20 July 2022
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	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 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!

	~~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~~.		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.

	~~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~~.		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.

	~~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~~.		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.

	~~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~~.		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.

	~~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~~.		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.

	~~== Rule~~ of ~~28 ==~~		Never ever assume you'll win a track, even if you only need to move one more, especially near the middle of the board.

	A researcher found an heuristic to help playing the game (Michael Keller, 1986). Each column has a number of points :		== The Rule of 28 ==
	* 2 and 12 are worth 6 points		A researcher found an heuristic to help playing the game (Michael Keller, 1986). Each column has a number of points:
	* 3 and 11 are worth 5		* 2 and 12 are worth 6 points
	* 4 and 10 are worth 4		* 3 and 11 are worth 5
	* 5 and 9 are worth 3		* 4 and 10 are worth 4
	* 6 and 8 are worth 2		* 5 and 9 are worth 3
	* 7 is worth 1		* 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.		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.		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.		Remember this strategy is not flawless.
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	http://www.solitairelaboratory.com/cantstop.html		http://www.solitairelaboratory.com/cantstop.html

	==Probabilities==		== Probabilities ==

	===Probability of ~~getting~~ a ~~particular number~~===		=== Probability of Getting a Particular Number ===

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

	===Combinations of ~~temporary markers~~===		=== 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.		On the first roll its 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
	(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):		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 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 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.		* 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.

	{\|style="width:auto; background-color:#f8f9fa; color:#222; margin:1em 0; border:1px solid #a2a9b1; border-collapse:collapse; text-align:right;"		{\|style="width:auto; background-color:#f8f9fa; color:#222; margin:1em 0; border:1px solid #a2a9b1; border-collapse:collapse; text-align:right;"
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	Some conclusions:		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)		* 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!		* 5-7-8 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)		* 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)		* 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).

Flo0: add probability tables

2021-01-24T18:09:35Z

add probability tables

← Older revision		Revision as of 20:09, 24 January 2021
Line 28:		Line 28:

	http://www.solitairelaboratory.com/cantstop.html		http://www.solitairelaboratory.com/cantstop.html

			==Probabilities==

			===Probability of getting a particular number===

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

			{\|style="width:auto; background-color:#f8f9fa; color:#222; margin:1em 0; border:1px solid #a2a9b1; border-collapse:collapse; text-align:right;"
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Number
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Probability
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|2 / 12
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|13.2%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|3 / 11
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|23.3%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|4 / 10
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|35.6%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|5 / 9
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|44.8%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|6 / 8
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|56.1%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|7
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|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.

			{\|style="width:auto; background-color:#f8f9fa; color:#222; margin:1em 0; border:1px solid #a2a9b1; border-collapse:collapse; text-align:right;"
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Chosen Columns
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Probability to continue
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Expected progress
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Breakeven progress
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[6, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|92.0%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.115
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.44
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|91.4%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.119
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.39
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 6, 7]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|88.7%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.113
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.999
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 7, 9]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|85.3%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.116
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.788
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 6, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|89.5%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.12
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.14
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[4, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|90.3%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.123
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.27
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[4, 6, 7]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|88.6%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.12
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.05
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[3, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|89.3%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.125
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.17
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[2, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|89.0%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.13
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.19
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[2, 6, 7]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|86.4%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.126
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.926
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[3, 7, 11]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|77.6%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.13
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.581
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[2, 7, 12]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|78.1%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.137
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|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)

Kevan: restore deleted section

2020-12-04T21:07:25Z

restore deleted section

← Older revision		Revision as of 23:07, 4 December 2020
Line 10:		Line 10:

	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.		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.

			http://www.solitairelaboratory.com/cantstop.html

Hammythepiper: /* Rule of 28 */

2020-12-04T20:34:16Z

Rule of 28

← Older revision		Revision as of 22:34, 4 December 2020
Line 10:		Line 10:

	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.		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.~~

	~~http://www.solitairelaboratory.com/cantstop.html~~

Sheffield refugee: Standardize the spelling for token; correct spelling/typing errors; correct plural/verb agreement

2020-09-20T04:13:43Z

Standardize the spelling for token; correct spelling/typing errors; correct plural/verb agreement

← Older revision		Revision as of 06:13, 20 September 2020
Line 13:		Line 13:
	= Rule of 28 =		= Rule of 28 =

	A ~~reasercher~~ found an heuristic to help playing the game (Michael Keller, 1986). Each column has a number of points :		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		* 2 and 12 are worth 6 points
	* 3 and 11 are worth 5		* 3 and 11 are worth 5
Line 21:		Line 21:
	* 7 is worth 1		* 7 is worth 1

	When you progress a black ~~tocken~~ on a column, add the number of points to your total. When you put a black ~~tocken~~ on the table for the first time, double the value. Whenever your total is equal or more than 28, stop.		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 ~~token~~ are on odd columns, and ~~substract~~ 2 if they are on even ~~column~~. Add 4 points if all the ~~token~~ are on value less than 8, or if all three are on values greater than 6.		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.		Remember this strategy is not flawless.

	http://www.solitairelaboratory.com/cantstop.html		http://www.solitairelaboratory.com/cantstop.html

Videofreak at 15:02, 3 September 2020

2020-09-03T15:02:17Z

← Older revision		Revision as of 17:02, 3 September 2020
Line 24:		Line 24:

	For further refinement, add 2 if all the black token are on odd columns, and substract 2 if they are on even column. Add 4 points if all the token are on value less than 8, or if all three are on values greater than 6.		For further refinement, add 2 if all the black token are on odd columns, and substract 2 if they are on even column. Add 4 points if all the token are on value less than 8, or if all three are on values greater than 6.

			Remember this strategy is not flawless.

	http://www.solitairelaboratory.com/cantstop.html		http://www.solitairelaboratory.com/cantstop.html

Heziva at 19:05, 14 August 2020

2020-08-14T19:05:20Z

← Older revision		Revision as of 21:05, 14 August 2020
Line 10:		Line 10:

	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.		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 reasercher 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 tocken on a column, add the number of points to your total. When you put a black tocken 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 token are on odd columns, and substract 2 if they are on even column. Add 4 points if all the token are on value less than 8, or if all three are on values greater than 6.

			http://www.solitairelaboratory.com/cantstop.html

← Older revision		Revision as of 20:09, 24 January 2021
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	http://www.solitairelaboratory.com/cantstop.html		http://www.solitairelaboratory.com/cantstop.html

			==Probabilities==

			===Probability of getting a particular number===

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

			{\|style="width:auto; background-color:#f8f9fa; color:#222; margin:1em 0; border:1px solid #a2a9b1; border-collapse:collapse; text-align:right;"
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Number
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Probability
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|2 / 12
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|13.2%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|3 / 11
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|23.3%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|4 / 10
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|35.6%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|5 / 9
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|44.8%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|6 / 8
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|56.1%
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|7
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|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.

			{\|style="width:auto; background-color:#f8f9fa; color:#222; margin:1em 0; border:1px solid #a2a9b1; border-collapse:collapse; text-align:right;"
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Chosen Columns
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Probability to continue
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Expected progress
			!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Breakeven progress
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[6, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|92.0%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.115
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.44
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|91.4%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.119
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.39
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 6, 7]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|88.7%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.113
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.999
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 7, 9]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|85.3%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.116
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.788
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[5, 6, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|89.5%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.12
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.14
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[4, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|90.3%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.123
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.27
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[4, 6, 7]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|88.6%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.12
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.05
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[3, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|89.3%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.125
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.17
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[2, 7, 8]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|89.0%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.13
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|1.19
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[2, 6, 7]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|86.4%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.126
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.926
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[3, 7, 11]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|77.6%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.13
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.581
			\|-
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|[2, 7, 12]
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|78.1%
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|0.137
			\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|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)

← Older revision		Revision as of 07:12, 20 July 2022
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	!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Probability		!style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em; background-color:#eaecf0; text-align:center;"\|Probability
	\|-		\|-
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|2 / 12		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|2 and 12
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|13.2%		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|13.2%
	\|-		\|-
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|3 / 11		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|3 and 11
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|23.3%		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|23.3%
	\|-		\|-
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|4 / 10		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|4 and 10
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|35.6%		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|35.6%
	\|-		\|-
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|5 / 9		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|5 and 9
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|44.8%		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|44.8%
	\|-		\|-
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|6 / 8		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|6 and 8
	\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|56.1%		\|style="border: 1px solid #a2a9b1; padding: 0.2em 0.4em;"\|56.1%
	\|-		\|-
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	Some conclusions:		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)		* 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.		* 5-7-8 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)		* 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).		* 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).