# 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.

• 2 tiles pays 4 times the stake
• 3 tiles pays 9 times the stake
• 4 tiles pays 16 times the stake
• 5 tiles pays 25 times the stake
• 6 tiles pays 36 times the stake
• 7 tiles pays 49 times the stake
• 8 tiles pays 64 times the stake
• 9 tiles pays 81 times the stake

## 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:

• [0-0] [5-6] [4-6]
• [0-0] [5-6] [5-5]
• [0-0] [6-6] [3-6]
• [0-0] [6-6] [4-5]

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:

[1-5] [4-4] [1-1] [0-6]

which has eight different ways of being scored:

[6] [4] [1] [6]   = 17
[6] [4] [2] [6]   = 18
[6] [8] [1] [6]   = 21  <= winner
[6] [8] [2] [6]   = 22   busted!
[6] [4] [1] [13] = 24   busted!
[6] [4] [2] [13] = 25   busted!
[6] [8] [1] [13] = 28   busted!
[6] [8] [2] [13] = 29   busted!

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

120000000
241000000
385000000
41311200000
52124700000
627422110000
737714470000
84510790220000
952157161543000
105921426812412000
116428241824241000
1265354626442104300
13674318927482371400
1464503122212134784900
15605751618185490013710
1655630207427401575323100
1748679257138992621695370
184070531045369416413571190
193372036417165636224873113
2026711417393039379429572216
2119691466511756133907113150065

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

[0-0] [0-1] [0-2] [0-3] [1-1] [1-2] [3-3]

with one of these tiles: [0-4], [1-3], [4-4]

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
```