A major part of the strategy of Can't Stop is when to stop rolling.  The point at which one should stop depends on the three numbers marked and the amount of progress already made in the current turn (measured in fractions of the total distance needed to win a column). The ideal stopping point can be calculated from the probability of a successful roll, and the average progress value of a successful roll.  The ideal strategy would consist of a table of 165 entries, one for each combination of three numbers, each entry showing the level of progress at which to stop for that combination.

Such a table would, of course, be impractical to use in an actual game. What we really need is a rule of thumb that is simple to use and reasonably accurate. The following rule, called the Rule of 28, should prove helpful. Add the following values for each number marked or advanced in each column during the current turn (note that each column counts double when a marker is placed).

Column               2  3  4  5  6  7  8  9 10 11 12
Spaces               3  5  7  9 11 13  9  7  5  3  1
Value When Marked   12 10  8  6  4  2  4  6  8 10 12
Value When Advanced  6  5  4  3  2  1  2  3  4  5  6

Note that each column counts double when a marker is placed. Add two points if all three columns marked are odd; subtract two points if all three columns are even. Add four points if all three columns are less than eight, or if all three are greater than six. When the total for the current turn reaches 28 or more, stop rolling (but do not stop before all three markers are placed).
    A few turns of a sample game:
(1) 1-1-3-4  Mark 2 and 7. Neither has been marked this turn, so count 12 plus 2, or 14.
     1-1-4-6  Advance marker 2, mark column 10. Add 6 and 8 for a total count of 28. This is the total at which to stop.
(2) 1-2-5-6  Mark and advance 7. Count 2 plus 1 for a total of 3.
     1-1-3-6  Mark 2 and 9. Add 12 plus 6 for a total of 21. Continue.
     1-5-5-6  Advance 7. The total is now 22. Continue.
     1-2-4-5  Advance 9. The total is now 25. Continue.
     1-3-3-5  Miss. Bad luck!
(3) 2-2-3-3  Mark and advance 5. Count 6 plus 3 for 9.
     1-2-2-4  Mark 3 and 6. Add 10 plus 4 for 23, plus 4 (because 3, 5, and 6 are all less than 7) for 27. Roll once more.
     2-4-6-6  Advance 6. Add 2 for a total of 29, and stop.

The Rule of 28 will tell you within one roll the correct point at which to stop in 75 percent of all cases (and within two rolls in 92 percent of all cases). This is a fairly good result for a simple, practical rule. However, since you are playing against opponents, do not follow the rule slavishly. Use your judgment as to where you stand in the game when making decisions on when to stop. In particular, when you are behind in the game, or are close to winning a closely contested battle for a column, it is a good idea to take extra chances (thus you would continue to roll even when your turn total exceeds 28).

Thanks to Robert Abbott for introducing me to this excellent game, and for calculating the table of success probabilities which led to this study.

Can't Stop was developed from an earlier Sid Sackson game, Sid Sackson's Solitaire Dice, which appeared in A Gamut Of Games.  This was designed as a solitaire game, but can also be played competitively.  This is also an interesting dice game, giving wide scope for players to exercise skill in choosing what combinations to try for and how to score each roll.  Briefly described, the player rolls five dice, and divides the roll into two pairs and a single die.  The scoresheet is labeled with the two-dice totals from 2 to 12, and the single die rolls 1 to 6.  The two pair totals from each roll are tallied off on the 2 to 12 section, and the single die on the 1 to 6 section.  The player continues rolling and scoring, but can mark off only three different numbers as singles.  The game ends when he has marked off a number as a single eight times. He then scores for each two-dice total he has marked off more than five times: 100 points per extra 2 or 12, 70 for each extra 3 or 11, 60 for each extra 4 or 10, 50 for each extra 5 or 9, 40 for each extra 6 or 8, and 30 for each extra 7.  He loses 200 points for each two-dice total he has used between one and four times.  This game is an excellent one for a duplicate format because there are no rethrows.    

A sample three-handed game of  'Duplicate Solitaire' Dice (The three numbers under each player indicate the two two-dice totals and the single die left over):

Roll     Alpha      Beta     Gamma    
22244    8  4 2    6  6 2    6  6 2   
15566   10 12 1   11 11 1   10  7 6   
11234    6  4 1    5  5 1    7  3 1  
12344    8  4 2    5  5 4    7  6 1    
13456   10  8 1   11  7 1    7  6 6  
12456    8  6 4   11  6 1   10  7 1   
11112    2  2 2    2  2 2    3  2 1   
11236    9  2 2    9  2 2    9  3 1   
12356    9  6 2   11  5 1    9  6 2   
12256    8  7 1    7  7 2    7  3 6   
23456    8  8 4    9  7 4    9  9 2
13666   12  9 1   12  9 1    9  7 6
14445    8  6 4    9  5 4    6  3 1
23446    9  6 4    9  6 4      out
14466   10 10 1   12  5 4      ---
11233    4  4 2    6  2 2      
11234    7  2 2    7  2 2      
23456    8  8 4   11  5 4
14566   12  6 4   11  7 4
11223    4  4 1    5  2 2
12346    8  4 4    9  6 1
22334    6  4 4    5  5 4

Alpha scored 360 for his 4's,  6's, and 8's, but took 1000 points in penalties (2's,  7's,  9's,  10's,  and 12's) for -640.  Beta played an excellent game, scoring 560 for his 2's,  5's,  6's,  7's,  9's,  and 11's, and took only 200 points in penalties (12's) to finish +360 and win.  Gamma was runner-up at -420 (180 for 6's, 7's and 9's, 600 penalty points for 2's, 8's and 10's).

Solitaire Dice has been published several times commercially: as Choice/Einstein (1989, Hexagames), Extra! (2011, Schmidt Spiele), Can't Stop Express (2017,  Eagle-Gryphon Games; this had a terrible rule change which was later reversed)

Another version appeared in Games Magazine (Jan/Feb 1978) under the title Great Races.  This was previously published in 1974 as part of The 6 Pack of Paper & Pencil Games (Gamut of Games), and later as Mother Road (2022, Eagle-Gryphon Games).

Ten Thousand Games to Play with Dice

One of the most popular groups of dice games are point-scoring games in which players have the option to end their turn at a given point, or risk the score already achieved on the turn by throwing further.  Perhaps the best-known of these is the French game of Dix Mille (Ten Thousand), so called because the object of the game is to score 10,000 points. This is a folk game, spread mostly by word of mouth, with many variants and many different names.   I originally learned the game as Sparkle, under which name I wrote about it in WGR4 and WGR5.  It appears in relatively few games compendia, even those devoted to dice games (see the Bibliography for some written references).  The appendix (Short Reviews of Games in Print) to the 1969 edition of Sid Sackson's A Gamut of Games lists a version by Parker Brothers under the name Five Thousand, a name sometimes used in English-language sources for a version of the folk game with a shorter goal of 5,000 points.

                     Scoring in Ten Thousand

1 = 100 points             1-1-1 = 1000 points   4-4-4 = 400 points
5 = 50 points              2-2-2 = 200 points    5-5-5 = 500 points
1-2-3-4-5-6 = 1500 points  3-3-3 = 300 points    6-6-6 = 600 points

Let's look at the basic rules, and some statistical strategy, before examining some commercial variants.  Ten Thousand is played with six dice. The dice are rolled and various combinations of dice score (summarized above):  Every one thrown scores 100 points; every five scores 50 points; every triplet scores 100 times the value of the number thrown (222 = 200, 333 = 300, etc.) except that 111 = 1000; and the sequence 123456 scores 1500.  After each throw, a player may stop and score all of the points accumulated on that turn (turn score) to her permanent score (game score).  Or she may roll any non-scoring dice again to try to score more points; any throw with no scoring combinations (zilch) ends the turn and wipes out her turn score, with nothing added to her permanent score on that turn.  If all of the remaining dice score, she may roll all six dice again and continue to add to her score.  The game is usually played in rounds; when one player reaches 10,000 points, every other player who has not rolled in that round gets one more chance to pass the leader.  This is a very interesting game which can be played solitaire or competitively, and which contains a fair amount of skill. Here are a few sample turns:

Roll   Score Turn  Score Total   Score Comments
124556   200  200      0         Must continue rolling; rethrow 246.
124      100  300
35        50  350
5         50  400                Rethrow all six dice.
135666   750 1150   1150         Puts the score well over 500. A good place to end the turn.

233566    50   50
33455    100  150
124      100  250   1400         Let's stop again.

123566   150  150
2556     100  250                Let's try for more this time.
16       100  350                Take another chance.
6          0    0   1400         Oops -- not as lucky this time.

Many variants of the game require a minimum game score (typically 500) before a player is allowed to stop voluntarily; some even require a minimum turn score on every turn before a player is allowed to stop. I dislike both rules: the skill of the game is in deciding when to stop; anything which reduces the number of decisions increases the luck factor. The minimum game score requirement also may put players hopelessly ahead or behind (if one player gets 500 on the first turn, and everyone else misses for two turns, she has such an advantage she may never be caught). An excellent variant rule is described by Peter Arnold in The Book of Games (under the game's original name, Dix Mille): a player reduced to one die has two throws to score (and a 500 point bonus for succeeding), increasing the chance of a rollover and rewarding more aggressive play.

Technical Analysis -- Ten Thousand
[Originally published in WGR5]

From an analysis of the various dice combinations, we can compile the following tables of probabilities and scores:

Number       Number of combinations           Total scores for each
of dice  for each number of dice thrown       number of combinations
left        6     5    4   3   2  1           6       5       4      3     2     1
  6      1548   192   48  12   4  2     1714950  112800   24450   8700   600   150
  5      7200     0    0   0   0  0      540000       0       0      0     0     0
  4     11520  2040    0   0   0  0     1728000  153000       0      0     0     0
  3     12132  2400  480   0   0  0     4369500  360000   36000      0     0     0
  2      8928  1704  384  96   0  0     4701600  615000   87600   7200     0     0
  1      3888   840  180  48  16  0     2329200  435000   65100   7200  1200     0
  0      1440   600  204  60  16  4           0       0       0      0     0     0

Total   46656  7776 1296 216  36  6

For example, when throwing four dice, 480 out of the 1296 possible throws will leave you with three dice, and these 480 throws score a total of 720 points. These tables were used to derive a basic strategy, which boils down to the question of what score is enough to stop and take a profit on the turn (keeping in mind that continuing to roll risks missing and losing your score for the turn). The score with which you should stop depends on how many dice you are about to throw. So our basic strategy will consist of a set of six stopping numbers. The stopping numbers for the general strategy (derived from formulas balancing potential adding gain from continuing against the risk of losing your total for the turn) are:

Number of dice to throw     6    5    4    3    2    1
Score on which to stop  10400 2950  850  250  150  150

For example, when you are left with four dice to throw, you will throw again if your turn score is less than 850, and stop if it is 850 or more.  Notice two important things about these numbers.  First of all, you will never throw one or two dice (except when required to make an initial score of 500).  By the time you reach one or two dice, your turn score will always be at least 200 (four fives).  Also, you will probably never stop with six dice to throw, and infrequently stop on five (a score a high as 2950 for a turn is very high, and the odds against ever reaching 10400 are astronomical). You will stop on three dice only when you throw 111, 333, 444, 555, or 666 on the first throw, when you throw three sixes in two or three throws, or when you have scored on all six dice (in one or more throws) and are on your second (or more) cycle (you will throw six dice a second time on about 6.6 percent of your turns).

This strategy gives the best average score per turn, and is suitable for use when playing solitaire or when playing competitively in many situations. It requires patience, however; you will stop after a single throw over 35 percent of the time. There are two situations where this basic strategy might be modified, however. The first is when you have not scored the necessary 500 points on a turn to begin accumulating points. Should you stop the moment you reach 500?  The second situation is in a competitive game when you are trailing by a wide margin.  How wide a margin requires a change in strategy?

Let's look at a few statistics obtained from a combination of probablility calculations, plus a computer program which played 100000 turns following the basic strategy, and 100000 turns trying to score 500 points.  The approximate probablity of scoring 500 points in a turn is 46.7%.  The computer scored 500 or more points in 46718 out of 100000 turns.  The total score in the 46718 scoring turns was 3,438,8150, an average of 736.08 points per success, or 343.88 points per overall turn.  The computer scored in 86264 out of 100000 turns when stopping according to the basic stopping numbers, fairly close to the calculated probability of 84.1%.  The total score for 100000 turns was 41,942,700 (419.427 per turn).

The results of this Monte Carlo simulation paint a clear picture of the risk you take if you decide to risk another throw after you have scored your first 10 points. Since slightly less than half of all turns needing a score of 500 are successful, you do not risk much by continuing to follow the stopping numbers on a turn in which you have reached your initial 500 points. Naturally this means you will stop if you have three or fewer dice to roll, and you will essentially always roll five or six dice. Perhaps you might wish to play it safe and stop on four dice -- unless all or most of your opponents have scored their first 500 points and you are willing to take a risk to try and catch up.  Strategy in situations where you are way behind is extremely difficult to quantify, and this is perhaps where the judgment of the player should take over.  There may even be situations where you want to roll one die in attempt to close the gap between you and the frontrunner.  Ten Thousand can provide some exciting play in a competitive game between a number of players.

Commercial versions of Five or Ten Thousand

Ten Thousand is perhaps the most popular dice game ever in commercial adaptations: even Yacht does not appear in as many different commercial versions (looking through flyers from Toy Fair and elsewhere, I see versions called Bupkis, Bummer, Farkle, etc).  In WGR8 we panned a variant called Fill or Bust! (Bowman, 1981); to my surprise this reappeared in a very similar vein a few years ago as Volle Lotte! from the German company Abacus, and unaccountably has received some positive press.  Many other versions have appeared since under different names, many with gimmicks such as jokers or special bonuses for reaching certain scoring totals exactly.     Some games add bonuses for three pairs or double triplets.

Bone Didley, n.d., Carjulin Corporation, Calgary
Cosmic Wimpout, designed by W. W. Swilling and the Boston Logical Society, published 1984 by Cosmic Wimpout, $3.50 for basic set, up to $16 for deluxe set in carrying pouch
Fill or Bust, inventor uncredited, published by Bowman Enterprises, 1981, $4.95
Gold Train, designed by Jeffrey L. Strunk, published 1995 by Strunk Games, $24
Greed$, published 1980 by the Great American Greed Company, distributed 1986 by Avalon Hill, $14.95 in tube
High Rollers, published 1992 by El Rancho Escondido Enterprises, $7.95 postpaid
Six Cubes, published 1994 by Fun and Games, $9.95 retail
Volle Lotte!, published 1994 by Abacus, about $12 as import
Zilch, published 1980 by Twinson Company, $6.50 from Miles Kimball gift catalog

Cosmic Wimpout is the most popular of the commercial Five Thousand variants; it even has its own Internet newsgroup and several World Wide Web pages devoted to the game. It is unusual in using only five (special) dice instead of six -- it also has a joker and several special rolls (e.g. the equivalent of 5-5-5-5-5 wins automatically, while 1-1-1-1-1 loses automatically). I don't find it an improvement on the standard Five Thousand myself -- there are too many rules limiting players' actions.

For players of Cosmic Wimpout (does anyone know anything concrete about the origin of Cosmic Wimpout?) who wish to adopt an analogous strategy, many of the above comments are applicable. Two major differences are the use of only five dice and the presence of a black die which contains a wild card. This requires the use of separate stopping numbers for dice totals with and without the wild card die (W indicates dice totals with the wild card). Also, a total of only 7 points is needed to begin accumulating a score. Certain rules, such as the Reroll Clause, have been disregarded, as they produce extreme complications in the calculations. The basic strategy for Cosmic Wimpout is (using the scoring table where the basic score is 5):

Number of dice    5  4W  4 3W  3 2W  2 1W 1
Stopping number 390 110 80 35 20 15 10 15 5

Fill or Bust! adds chance cards to Five Thousand, which adds a great deal of luck, reduces the number of options the player has, and makes many of the remaining decisions trivial -- the little skill remaining is overwhelmed by the chance element.  It was republished in 1994 as Volle Lotte!

High Rollers uses a set of dice in which every number (1-6) has different colored pips on each die.  The scoring system gives extra value to a straight (1-2-3-4-5-6) in which the six numbers are the same color or six different colors, but the dice were misdesigned so that a six-colored straight is impossible!  The dice don't add much to the basic game of Five Thousand, but they can be used to design highly interesting versions of Yacht, using four rolls per turn instead of three, and adding color-based categories (e.g. six-color throw) as well as number-based ones.

The standard version of Zilch is essentially the same game as Five Thousand, but the variant games add extra scoring throws (e.g. three pairs or two triplets) and extend the goal to 10,000 or 15,000 points -- this improves the game a bit by making the average throw (especially of six dice) more valuable and encouraging longer turns.  There is also a version in which throws of less than six dice can be combined with saved dice to increase the score.  Greed$, by Avalon Hill, uses specially marked dice but is essentially a straightforward version of Five Thousand.

Six Cubes, a recent board game version of Ten Thousand, uses four white dice plus a red die and a green die. Rolling a six on the red die (except as part of a scoring combination) ends your turn with no score; a green six doubles your points for the entire turn. The board has a 10,000 point track with three Incentive Squares: ending your turn on one earns a 1000 point bonus.  I like those, but not the colored dice, which increase the luck factor of the game.

The nicest physical production of the game is Strunk Games' new version Gold Train, which comes with a train-shaped wooden scoreboard (similar to a cribbage board) with a compartment to hold the dice and pegs. Gold Train is another five-dice variant (though using standard dice, unlike Cosmic Wimpout), adding bonuses for four and five of a kind and four and five dice straights, plus a rule which rewards aggressive play: a player who stops after surpassing the score of another player may attempt a Train Robbery (essentially a free turn with the points from the turn subtracted from the overtaken player).  There are also a few money bags on the pegboard, similar to the Incentive Squares in Six Cubes.  Two rules in Gold Train I dislike are Train Wreck: a 500 point penalty for a non-scoring throw with five dice, which encourages passive play, and End of the Line: you must reach 10,000 points by exact count, which adds too much luck to the finish of a close game. 

Pig is one of the simplest games of the jeopardy (push your luck) dice games.    A single die is used.  The object is to reach a total score of 100.  On each turn a player rolls the die, keeping a running total of the rolls, and may stop at any time and add the total for that turn to her permanent score.   Throwing a 1 ends the turn and wipes out the total for the turn.

Liar's (or Liar) Dice is a well-known dice game which is described in detail in various editions of Hoyle. 

Shut The Box and Tric-Trac are two of several names for a simple dice game which can be played solitaire or competitively.  It has appeared commercially as Snake Eyes. In its most common form, it consists of a box with markers numbered 1 through 9, which are flipped over to indicate that they are out of play. Two dice are thrown, and the player flips one or more numbers which add to the sum of the dice; e.g. on a throw of 10 (i.e., 5-5 or 6-4 -- the individual dice do not matter), combinations which can be flipped are 9&1, 8&2, 7&3, 6&4, 7&2&1, 6&3&1, 5&3&2, 5&4&1, or 4&3&2&1.  The object is to flip as many numbers as possible before throwing a total which cannot be covered.  A win is achieved by flipping all nine numbers.  Whenever desired (usually when all numbers larger than 6 are flipped), one die may be discarded for the rest of the game.  A quick sample (* throw of one die):

Throw     6-1     6-2    5-2   6-4  4-4   1*
Flip        7       8    4,3   9,1  6,2   --
Left 98654321 9654321  96521   652    5   (5)

Several scoring systems are possible -- e.g., you may count how many flippers are left or the total of their numbers.  The chance of winning with best play is about 8 percent.   A 1000-game trial produced 85 wins, leaving an average of 2.3 flippers, totalling 11.43 points.   The worst score was 8 flippers left for a score of 43.   The game is occasionally seen with more than 9 flippers.   A ten-flipper game is playable (6 percent chance of a win), but the eleven- (1 percent) and twelve-flipper (1/2 percent) games are far too difficult to win.   I have played many games of a giant version with 8 dice and flippers numbered 1 to 30. More dice and flippers give more scope for skillful play.


The Simpsons "Don't Have A Cow" Dice Game (Milton Bradley, 1990)

Most commercial games named for T.V. shows, movies, musical groups, and the like are simple race games or other games of chance, not intended for serious (or even semi-serious) players.  It was nice to see that this neat little dice game, based on the popular animated T.V. program, has some interesting strategic features. It's an easy game to learn, and can be played as a family game: older players can help younger players select their bets carefully.   Basically it is a betting game -- the player whose turn it is to roll selects one of eight pictures (containing one or more of the Simpson family members), then decides how many dice to roll to try to match the picture, and how many chips to bet. If he is successful, he wins the same number of chips from the bank; if not, he loses his bet to the bank. Each of the other players bets either for or against the roller, winning or losing in the same manner. But the most important point is that if the roller is successful, he wins the chips from players who bet against him. This makes betting against the roller more dangerous, and encourages the roller to take greater chances to encourage players to bet against him. Another important rule is that the game ends when the bank or any player runs out of chips (the player with the most chips wins).

The roller has three main choices to make (as outlined above), and each non-roller has two choices (how many chips to bet, and whether to bet for or against). Several factors need to be considered when making selections, including the number of chips each player has, and the number of chips left in the bank. A player in the lead wants to try to maintain her lead while bringing about the end of the game, while trailing players want to prolong the game while attacking the leader. When the leader is on roll, it will often be a good idea to bet for her to avoid giving her extra chips; conversely, it may be wise to bet against a player in danger of running out of chips, to keep him from running out.

In order to play well, it is useful to know what the probabilities of each roll are. The table below gives the five essentially different pictures, and the probability of achieving each picture with the allowable numbers of dice, rounded to three decimal places.

Number of People  Number of Dice Thrown
 in Picture     1    2    3    4    5    6    7   8
    1         .333 .556 .704
    2              .194 .407 .584
    3                   .157 .363 .549
    4                        .036 .115 .224 .344 .463
    5                             .199 .423 .607 .740

Gemstones (Mayfair Games, 1987, $3)

Gemstones is a 'free' game given away with a reasonably priced package of 5 polyhedral dice (4-, 8-, 10-, 12-, and 20-sided).  In each round each player has five throws and must save exactly one die on each throw; the object being to get the highest total of the five dice (plus bonuses for pairs or better, straights, etc.).  The strategy is too clear-cut; a better rule would be to take five throws, saving as many or as few dice each round as desired, Yacht-style; dice thrown a fifth time must be kept.  This leaves more room for decision making and increases the chance of bonuses.  The bonuses themselves are too small, in comparison to both the main scores and the difficulty of achieving them.  For example, a 5-point bonus is given for throwing five 1's on the first throw. This should happen once in 76800 throws, and the 5-point bonus is small compared to an average score of slightly over 41 points per round which can be achieved with a simple strategy.  Worse yet, the bonuses which can be usefully pursued (pairs, all odd, all even) will not help produce high scores.  The answer may be to greatly increase the bonuses for pairs.

Basic strategy table for Gemstones
Keep a die after a particular throw if its value is at least that shown on the table. For example, keep the 10-sided die after the third throw if 8 or higher is thrown.

     Throw
 Die  1  2  3  4
   4  4  4  3  3
   8  7  7  6  5
  10  9  8  8  6
  12 11 10  9  7
  20 17 16 15 11

Uno Cubes
Don't Go to Jail (Parker Brothers, 1991) (invented by Garrett J. Donner and Michael Steer)
Dice Deck

The following table of dice probabilities may be useful in a variety of dice games in which dice totals, rather than combinations, are the important factor.  John Scarne's book has similar tables going up to five dice, but it contains two incorrect figures for four dice totals.

     Numbers of Combinations Of Various Totals With Two To Six Normal Dice
     (Numbers in parentheses indicate the total number of combinations)

   Two Dice (36)  Three Dice (216)   Four (1296)   Five (7776)    Six (46656)
   2, 12    1       3, 18    1       4, 24    1    5, 30    1    6, 36     1
   3, 11    2       4, 17    3       5, 23    4    6, 29    5    7, 35     6
   4, 10    3       5, 16    6       6, 22   10    7, 28   15    8, 34    21
   5,  9    4       6, 15   10       7, 21   20    8, 27   35    9, 33    56
   6,  8    5       7, 14   15       8, 20   35    9, 26   70   10, 32   126
       7    6       8, 13   21       9, 19   56   10, 25  126   11, 31   252
                    9, 12   25      10, 18   80   11, 24  205   12, 30   456
                   10, 11   27      11, 17  104   12, 23  305   13, 29   756
                                    12, 16  125   13, 22  420   14, 28  1161
                                    13, 14  140   14, 21  540   15, 27  1666
                                        15  146   15, 20  651   16, 26  2247
                                                  16, 19  735   17, 25  2856
                                                  17, 18  780   18, 24  3431
                                                                19, 23  3906
                                                                20, 22  4221
                                                                    21  4332

Widely used in fantasy games like Dungeons & Dragons, dice are now being widely manufactured with the following numbers of sides: 4, 6, 8, 10, 12, 20, 24, 30, and 60.   Other numbers have been made but are much less common (except for two sided dice, which are usually coins).  Dice with 4, 6, 8, 12, and 20 sides use the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron).   Dice with a number of sides twice a prime can be made by joining two pyramids with a regular polygonal base (10-sided dice use two pentagonal pyramids, sometimes offset by half a side to produce kite-shaped faces). Attaching a shallow square pyramid to each side of a cube produces a 24-sided die.  Prime-numbered dice can be made as long prisms, or by repeating numbers on a larger die (a game which needs results from 1-5 could use an icosahedron with each number repeated four times).   Photos of many different dice can be found on Wikimedia Commons.  

The first game I saw with 12-sided (dodecahedral) dice was a simple dice baseball game called Dodeca, which used six special dice labeled with outcomes for various baseball events (including three different dice to represent three levels of hitting ability).  The popular baseball simulation Strat-O-Matic, which for years used a Split Card deck of 20 numbered cards to produce fielding and other subsidiary results, switched over to 20-sided icosahedral dice when they became widely available.

Although 100-sided dice have been made (e.g. Lou Zocchi's Zocchihedron, which resembles a golf ball with 100 tiny round sides), some sports and fantasy games use a special pair of ten-sided dice, one die labelled by tens (0-10-20-30-40-50-60-70-80-90) and the other by ones (0-1-2-3-4-5-6-7-8-9) to produce a set of possible outcomes from 0-99 (or 1-100 if 00/0 is read as 100).  

Dice other than the standard 6-sided ones can be used to create variants of well-known dice games. For example, craps could be played with other pairs of dice. However, in order to keep the odds of winning approximately 50-50, it is necessary to change the numbers which win at once (7 and 11 in standard craps), lose at once (2, 3, 12), and point numbers (4-6, 8-10).  Interesting variants of Yacht, Five Thousand, and Solitaire Dice or Can't Stop could also be played with dice other than the standard ones.  For example, 8-sided Solitaire Dice could be played, allowing four instead of three single numbers to be used, and using the following scoring values:

100  80  70  60  50  40  30  20
  2   3   4   5   6   7   8   9
 16  15  14  13  12  11  10

Chase -- invented by Tom Kruszewski (reviewed in WGR7, p23)
Duell -- Lakeside, 1976
Freez! -- Judith Lahore, Freez Games, 1993
Genius Attack -- Invicta
6-Dice -- invented by Jr. A.N. Verveen, 1988, Leiden, The Netherlands
   10x10 board, 10 dice per side (2-handed), 6 dice per side (4-handed)
Tonobeb -- invented by Bruce Hollock, published in Gameplay magazine, 1:1, Feb 1983
Moonstar -- invented by Alex Randolph, published 1981 by Avalon Hill (based on the earlier Corona, published 1974 by Kosmos; later published under other names) 

Domino Puzzles

Quadrilles -- Édouard Lucas

Multiplication Problems
   This is a multiplication problem rendered in dominoes.  So far I have not been able to find a 5x5 solution.

The Mechanisms of Domino Games

Initial distribution of tiles
Most games are for two to four players, either individually (round) or in two partnerships.   The tiles are placed face down on the table and mixed.   Players initially draw an equal number of tiles, depending on the game, with remaining tiles, if any, forming a boneyard (equivalent to a stock in card games).   In block games, there is no boneyard; either all of the tiles are distributed, or those not drawn are out of play for the whole deal.

Objective
The normal objective in the large family of block and draw games is to go out (play all your tiles before the opponents, also called going domino).   In point games, another objective is to score points by creating certain patterns on the board (e.g. in Muggins/Fives, to make a play that makes the sum of the free ends a multiple of 5, or in Bergen, to match two or more free ends).

Initial Play
Usually the first play (e.g. Block or Fortress) is made by the player holding the highest double; less often the tile with the largest pip count (referred to as heaviest).   Some games instead draw for first play at the start (each player draws one tile: the heaviest goes first, and the drawn tiles are returned and remixed before players draw their initial tiles), and the first player can start with any tile.  In some games the largest double in the set is played (by anyone) before players select tiles.   Play is almost always in clockwise rotation (to the left). 

Scoring
In the most common scoring system, the player who goes out first scores the total of all the pips on the tiles still held by the opponents.  If all players are blocked, the player holding the lightest hand (fewest total pips, or lighest individual tile in the case of a tie) scores the total of all opponents minus their own total.  A game may be played to a target score, or for a fixed number of deals (rounds).   

See the description of All Fives and Bergen below, the two most common point games, in which points can be scored in play, as well as at the end of each deal.

Tricktaking Games
The tiles are divided into suits, with doubles belonging to one suit and non-doubles belonging to two.

The most popular is Texas 42, a four-handed partnership bidding game: this is a complex tricktaking game in which 6-4 and 5-5 score ten points each; 5-0, 4-1, and 3-2 score five points each, and each trick taken scores 1 point, for a total of 42 points per deal.   See the Bibliography for several books on the strategy of bidding and play.

Solitaires
Most domino solitaire games are adaptations of card solitaires; packing is sometimes replaced by normal edge matching.

Summaries of the most common block and draw games

Block
The simplest block game, played with 2 to 4 players with a double-six set.   In the two-handed game, each player draws 7 tiles (5 each in three- and four-handed).  The player with the highest double plays first.  There is no drawing; a player who cannot play passes.    Doubles are usually played sideways, but are not spinners; they can only be matched on the long edge, so there are always two ends in play.   Score as usual when a player goes out, or all four players pass.


A blocked position with only 12 dominoes played

Fortress
A simple block game (unrelated to the well-known card solitaire) with a double-six or double-nine set, and 3-6 players.  In the three-handed (double-six) game, each player draws 9 tiles.  The player with the 6-6 plays first, then draws the last tile (so the boneyard is empty).    Doubles are played sideways, and are spinners: they can be matched on all four sides, so there can be many free ends.  A player who cannot play passes.  When a player goes out, score as normal.  Accounts of this game vary, and some sources seem to confuse it with Sebastopol (at least one source gives Sebastopol and Fortress as alternate names for the same game).  

Maltese Cross
An elaboration of Sebastopol, in which the four ends branching from the initial double must be matched with doubles on the next four plays, after which play continues normally on the four ends.    Players who cannot play draw one tile, which they must play if possible, or pass their turn otherwise.

Chicken Foot
Another game similar to Sebastopol, usually played with a double-nine (or double-twelve) set, and played in a series of games in which the first tile played is 9-9 in the first round, then decreases by 1 each round, with the tenth round starting with 0-0.   The intitial double must be matched four times before any other plays are made (as in Sebastopol).  Players who cannot play draw one tiles and play it or pass.   Each subsequent double played to one of the four lines must be matched three more times before any other plays are made: first in the middle of the long edge, then twice at 45 degrees, forming the shape that gives the game its name (picture above).   Scoring is negative, as in the card game Hearts: when a player goes out, they score zero for that deal (unless they went out with a double, which is a 50-point penalty), and each of the players scores the sum of the pips on their remaining tiles (but 0-0 counts 50 points), and the lowest score after ten rounds wins.   Some sources credit Louis and Betty Howsley with inventing Chicken Foot in 1986, but it seems to have many variations.

International Dominoes
A four-handed partnership version of Block, one of the most popular worldwide variants.  It is played two against two with each player drawing 7 tiles (so there is no boneyard).   If any player draws five or more doubles, they may call for a redeal (in some versions, this is mandatory).   Doubles are not spinners and there are only two ends in play.  A player who cannot play passes, but you must play if possible.  When a player goes out, their partnership scores the total of both opponents' tiles, disregarding the partner of the player who went out.   If a deal is blocked, the side with the lowest total scores the total of the opponents, regardless of their own total.   Most commonly the game is played to 100 points.  Rules for the first play vary: Anderson and Varuzza specify that the first deal is always started by the player with 6-6, who plays it.   First play rotates to the left on subsequent deals; the player playing first must start with a double if possible.  Lugo specifies that players draw to determine who starts the first deal, and allows any domino to be played.  

Pagat gives a number of different variants (some regional variants gives first play on each deal to the winner of the previous deal).   One set of international rules specifies drawing for first play, and allow any tile to be played first.  The winning team in each deal scores all of the unplayed pips (including the partner of a player who goes out), and the game is played to 200 points.

In the version called Milo, popular in Africa and Indonesia, the team winning each deal (by having a player go out or by having the lowest total if the game is blocked) scores one point, but if a player goes out and leaves both playable ends with the same number (cf. Bergen below), their team scores two points.  First team to 5 points wins.

All Fives (Muggins, Five-Up, Fives, Sniff)
Another of the most popular games is a scoring game, where any play which makes the sum of the playable ends a multiple of 5 scores that many points for the player who played it.  See the books by Armanino and Palmer for more details and strategy.   Any tile can be played to start; the first double played is a spinner, while subsequent doubles are not, so there are at most four playable ends.

In Fives and Threes, popular in Great Britain, any total which is a multiple of 3 scores 1 point for each three (3 points for 9, 4 for 12, etc.) and the same for multiples of 5 (2 points for 10, 4 for 20); a total of 15 scores 8 points (5 for five 3's and 3 for three 5's).

Bergen
There are various rules for this, particularly in how the winner of a blocked game is determined.   It is played by two to four players, drawing six tiles each (5 for four players).   Play starts with the lowest, not highest, double (or the lightest tile if no one has a double).   Doubles are not spinners; there are only two free ends.  A player who cannot play must draw until they can play, or until the boneyard is reduced to two tiles (which can never be drawn).   A play which makes both free ends show the same number (including the initial play of a double) is a doubleheader and scores 2 points.  Playing a double at either end of an existing doubleheader, or matching a double at the other end, is a tripleheader, and scores 3 points.  Going out scores 2 points, as does having the best hand in a blocked game (usually the lightest hand; several sources have unnecessarily complex rules involving doubles).  Some sources only give 1 point for winning a blocked hand; some also only give 1 for going out.   Usually played to 15 points for two players, or 10 points for three or four.   If you are 2 or 3 points from victory, you cannot win with a doubleheader or tripleheader (they only count 1 point each if you are two away, or 2 if you are three away, putting you 1 point shy in either case).

Puremco (now The American Domino Company), in business since 1954, is one of the world's leading domino manufacturers.   They make a variety of sets up to double-18, in many colors and styles, including personalized sets, plus cases, playing boards for some of the more recent variant games, and other accessories.

Cardinal Industries is a manufacturer of board, card, and dice games, including domino sets.

Dice Games

Arnold, Peter, editor -- The Book of Games, 1985, Exeter, 256 pp., ISBN 0-671-07732-5
Pages 79-80 cover Dix Mille.

Jacobs, Gil -- World's Best Dice Games, new edition, 1993, Hansen, 213 pp., paperback,  no ISBN, $6.95
Pages 80-90 cover Five Thousand (Zilch or Farkle) and Ten Thousand.  This is a revised edition of Jacobs' 1981 book, with a good bit of new material. Both editions contain descriptions of some folk variants of Liars Dice (mostly simplified versions, and none with the ingenious features of Richard Borg's game). The revised edition also contains rules for Dudo, a South American folk version which probably gave rise to Perudo (which is also mentioned briefly).

Mohr, Merilyn Simonds -- The Games Treasury, 1993, Chapters, 351 pp.[P], ISBN 1-881527-23-9, $19.95
Pages 103-104 cover Farkle.

Scarne, John -- Scarne on Dice, Eighth Revised Edition, 1980, Crown, 496 pp., hardback, ISBN 0-517-541246, $14.95
The bulk of the book covers Craps and other gambling games.  Yacht and its Puerto Rican cousin Generala are covered on pp. 364-368.   Scarney's own invention Scarney Dice is covered in Chapter 17 (pp. 370-394), followed inexplicably by three of his card game inventions.

Scarne, John -- Scarney Dice, 1969, John Scarne Games, 84 pp., spiralbound
40 dice games designed for a special set of dice with the word dead marked on two sides where 2 and 5 would normally be.

Vancura, Olaf -- Advantage Yahtzee, 2001, Huntington Press, 154 pp., paperback, ISBN 0-929712-04-8, $6.95
A computer analysis of Yahtzee, concluding that the average score with optimal play is 254.6.

"Spots Before The Ice", Games & Puzzles 54, November 1976, pp. 14-15 (Dix Mille)

The Cosmic Wimpout page, the company's official home page, includes various links, including an FAQ.

Armanino, Dominic C. -- Dominoes : Popular Games, Rules, and Strategy, 1977, Simon and Schuster, 176 pp., hardback (1987, Fireside, paperback)
  Survey of many standard games (including Five-Up, Seven Rock, Muggins, and Bergen), plus a variety of blocking, round, and bidding games.  Reprinted in 1978 by Sterling (128pp., hardback), omitting the chapter on bidding games.

Kelley, Jennifer A. -- Great Book of Domino Games, 1999, Sterling, 95 pp., paperback, ISBN 0-8069-4259-2, $6.95
  Collection of games for two or more players, including modern games like Mexican Train and Chickenfoot not found in older books.  Includes original games contributed by various inventors like David Galt, some of which use special tiles added to the regular set.  There is also an assortment of solitaire games, and some games using the 32-tile Chinese set.

King, Tom -- Popular Domino Games, n.d., Foulsham, 64pp., paperback
  Rules for 19 common domino games.   Bookseller lists give a publication date of 1950, but that may be a reprint; the same author's book on patience was published in 1920.

Lankford, Mary D. -- Dominoes Around the World, 1998, 40pp., paperback, ISBN 0-688-14051-3

Lewis, Victor T. -- Domino Games, 1971, 1980, Puremco, 56pp., paperback
   Rules for seventeen games, including matching/blocking games, bidding games, and children's games, plus a brief history, a section on general strategy, and a glossary.

Milton Bradley -- Dominoes, 1970, 2 pp. leaflet
Rules for five games, included with MB's Dragon Dominoes, one of the first double-12 sets.

Müller, Reiner F. -- Dominoes, Basic Rules & Variations, 1995, Sterling, 95pp., ISBN 0-8069-3880-3 (translation of Spielend Domino Lernen, 1987 [German])
Rules for about 19 standard domino games, plus puzzles, solitaires, and games for children.

Musante, Michael -- Pai Gow, Chinese Dominoes, 1981, GBC Press, 78pp., paperback, ISBN 0-89650-815-3
Description of a casino game played with the set of 32 Chinese dominoes.

Newsome, Travis -- Dominoes Game Night: 65 Classic Games to Entertain and Excite, 2023, Black Dog & Leventhal, 259pp., hardback, ISBN 978-0-7624-8123-1, $23.00
  An opening section on history and strategy, followed by a wide-ranging survey of games divided into bidding, blocking, Asian, scoring, and solitaire.   Curiously spends 13 pages on games played with a Cardomino set of 54 playing card dominoes, which could be used to play virtually any card game. 

Clarke, R.J. -- Domino Games, 50 Different Game Variations, 2016, Clarke, 109 pp., paperback, print on demand, ISBN 978-1530176151,
  Rules for both well-known and (probably) original games, with very brief strategy notes, illustrated in black and white.

Perkins, Bill -- Dominoes Plus, 2001, iUniverse, 236pp., paperback, ISBN 0-595-20576-3, $14.95
  Collection of over 100 games, mostly the author's own inventions.   Many games also use components from board games, dice, and playing cards.   The author prefers using a set of dominoes with all of the non-doubles repeated (so a double-9 set contains 100 tiles instead of 55).

Sackson, Sid -- A Gamut of Games, 1969, Castle Books, 1992, Dover, 210 pp., paperback, ISBN 0-486-27347-4, $6.95
  See the Board Games Bibliography for more about the book as a whole.  One of the original games by Sackson is The Domino Bead Game, inspired by Herman Hesse's Das Glasperlenspiel (also known as Magister Ludi).  This has a mechanism unlike any traditional domino games, and was generally well-received by the abstract game community.   The book is still in print from Dover.

Windham, Shane -- Table Games, 2013, Windham, 73 pp., paperback, ISBN 978-1475100785,
  Domino Games (pp. 39-48) describes eight of the author's original games.   No illustrations.

Puzzles (mostly)

Berndt, Frederick -- The Domino Book, 1974, Thomas Nelson, (Bantam, paperback), 195 pp.
Brief rules for 20 common games are given on pp. 7-36.  Ten solitaire games are described on pp. 37-51.  The remainder of the book covers puzzles, with solutions given for all.

Leeflang, K.W.H. -- Domino Games and Domino Puzzles, 1975, St. Martin's, 162 pp., hardback  (translation of Dominospelen en Dominopuzzels, 1972 [Dutch])
The first 26 pages give rules for a dozen or so games and variants.   The remainder of the book covers domino construction puzzles.  Some of the puzzles are magic squares,  others are arrangements of the double-six set so that half-dominoes with the same number are all in groups of four, either in a line, or in a square (the Quadrille puzzle named and studied by Edouard Lucas).

Planet Dexter -- Dominoes, 1996, Scholastic, 48 pp., paperback, ISBN 0-590-97224-3
Children's book on domino activities, including a few games.

Strategy Guides for Individual Games

Badum, Jody, and Richard Hay -- Texas42: Zero to Hero, 2023, Southpaws Playschool, 98 pp., paperback, ISBN 978-0-9978269-4-4, $9.95
Mack, Keith -- Texas 42 State Domino Game, Mack, 94 pp., paperback, ISBN 979-8-218-32902-0, $14.95
Roberson, Dennis -- Winning 42: Strategy & Lore of the National Game of Texas, 5th edition, 1997, 2000, 2004, 2008, 2020, Texas Tech University Press, 187pp., paperback, ISBN 978-1-68283-057-4, $18.95

Websites

McLeod, John -- Pagat
Huge collection of card games, including a page indexing rules for more than 150 domino games.

Spaans, Teun -- Domino Plaza
Long-running site with rules for many domino games and puzzles.

Portions of the section on Ten Thousand appeared in WGR4 (pp.8-9, February 1985), WGR5 (pp.23-24, September 1985), WGR8 (p.35, July 1988), and WGR13 (pp.22-23, February 1998).  The original version of the section on Can't Stop appeared in WGR6 (pp. 33-34, September 1986).

Most recently edited on August 31, 2024.  This article is copyright ©2024 by Michael Keller.  All rights reserved.