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Domino Twenty-One

Domino twenty-one is an adaptation of the card game 21 or Black Jack Straight using a double six set of dominoes.

Players and Equipment

The game is for two to five players with a double six set. With changes in the rules and a larger set of dominoes, the game can be extended to a greater number of players.

The Deal

Before dealing the hands, one player is designated as the banker and all bets are made against him. Each player stakes a predetermined amount of money or chips for each tile in his hand, the price of a tile, and the banker matches it with the square of the amount. That is, if the player puts up one penny per tile, he has to start with 2 cents to buy his initial hand, and the banker matches it with 4 cents. If the player buys another tile he pays another penny for a total of 3 cents, which the banker matches it with 9 cents, and so forth up to nine tiles. It is not possible to stay under 21 with a hand of more than nine tiles.

The Play

Each player draws two tiles after placing his original stake, regardless of the number of players in the game. The remaining tiles are left face down in a boneyard. At this point it is impossible for any player to exceed 21 points (see scoring below).

The banker goes clockwise around the table asking if each wants to stand (play the tiles he has) or take a hit (receive another tile). If the player elects to take a hit, he adds the price of another tile to his stake. He continues to takes as many hits as he wishes and when he decides to stand, the next player has the same options.

As each player receives or declines a tile, he must expose one of his three face down tiles on the table to the other players. This leaves him with two face down tiles after each purchase. All these tiles are still part of his hand.

The banker plays last and has the same options.

If a player exceeds 21 points, he must announce that he is busted, and expose all of his tiles. The banker wins his pot immediately.

The banker is the last to play and settles with each player separately. The bank rotates to the left after each hand.

Scoring

The value of a tile is

1) The number of pips, which gives us a score of 0 to 12

2) A double can be played according to rule one or scored as half its pips (i.e. [4-4] is either 4 or 8).

3) A blank half can be played as either 0 or 7 points (i.e. [0-4] is either 4 or (4+7)= 11 points).

4) Following these three rules, the [0-0] tile is worth 0, 7, or 14 points.

When the banker has settled on his hand, the hands are exposed and compared to the hand held by the banker. The banker can:

1) Announce that he went over 21 points (busted), not show his hand, and give each player his stake.

2) Announce his total and expose all the tiles in his hand. He will pay the players who beat his score and collect from the players who tied or scored lower than he did.

3) Any player who holds a three tile hand that equal 21 points which includes the [0-0] automatically wins over all other hands which score 21 points or less, including the banker's hand. Those four hands are:

Comments & Strategy

The later players have more and more information from the exposed tiles. Don't be the first player.

Since the double blank tile can only improve your payoff and never bust you, it is an important piece in the game. The possibility of getting one of the four special three-tile 21 hands also adds to its value.

It is hard to figure the score of your hand when it gets larger because of the many possible ways to compute blanks and doubles. For example, consider this hand:

which has eight different ways of being scored:

The strategic point to consider is if the risk of getting a fifth tile, which could bust you, is greater than the improved payoff for five tiles, namely 25 times your original stake.

In this example, you could count the hand as 17 and look for a tile that totals 4 or less. You already have the [4-4], so you would need the [0-4], [2-2] or [3-1] to have a winner. If you count the hand as 18, you need a tile that totals 3 or less. That would mean the [0-3], [3-3], or [1-2] would give you 21. You would be safe with the [0-0], [0-1] or [0-2, but not have a winner.

Look at the exposed tiles, determine which of these tiles are already in play, how many tiles are left in the boneyard and then make a decision. With each tile you add, the number of possible scores can double.

The number of possible hands of (n) tiles is easy to calculate as (28 * 27 * ...). Here is a table of the numbers of hands of (n) tiles which come to a given total between 1 and 21.

total23456789
1 2 0 0 0 0 0 0 0
2 4 1 0 0 0 0 0 0
3 8 5 0 0 0 0 0 0
4 13 11 2 0 0 0 0 0
5 21 24 7 0 0 0 0 0
6 27 42 21 1 0 0 0 0
7 37 71 44 7 0 0 0 0
8 45 107 90 22 0 0 0 0
9 52 157 161 54 3 0 0 0
10 59 214 268 124 12 0 0 0
11 64 282 418 242 41 0 0 0
12 65 354 626 442 104 3 0 0
13 67 431 892 748 237 14 0 0
14 64 503 1222 1213 478 49 0 0
15 60 575 1618 1854 900 137 1 0
16 55 630 2074 2740 1575 323 10 0
17 48 679 2571 3899 2621 695 37 0
18 40 705 3104 5369 4164 1357 119 0
19 33 720 3641 7165 6362 2487 311 3
20 26 711 4173 9303 9379 4295 722 16
21 19 691 4665 11756 13390 7113 1500 65

The three hands that total to 19 are the only nine tile hands in the game:

The thing that you will notice is that there is not a simple curve to the distributions and frequencies. Each new tile adds over 20 times as many possible hands 

tiles    Hands
========================
   2                756
   3             19,656 
   4            491,400
   5         11,793,600
   6        271,252,800
   7      5,967,561,600
   8      1.253187936 * 10^11
   9      2.506375872 * 10^12 

  tiles    odds of making 21
 ==============================                    
   2      .0251322751323
   3      .0351546601547
   4      .00949328449328
   5      .000996811830145
   6      .0000493635457404
   7      .00000119194412673
   8     1.19694736672 * 10^-08      
   9     2.59338596123 * 10^-11
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