Introduction

Blackjack is the most well-studied casino game. With a mathematic analysis, it has been shown to be the most profitable game (assuming good house rules, such as the Strip rules). In this article, we will briefly discuss how the game is played and what the correct strategy is in order to have a profit. Also, some counting systems and betting strategies will be introduced. A programming proof is shown at the end.

How the Blackjack is played

Blackjack game is a table game with a dealer standing on one side of the table and dealing cards to players, and all players (5 to 7 seats) sit on the other side of the table and play against the house. Every player is assigned a spot and his/her bet should be placed inside that spot. Both the dealer and player are dealt 2 cards after a bet is placed. One of the dealer's card, called upcard, is flipped; player's cards may be face-up or face-down. Based on the number of upcard and the total of player's cards, the player makes his/her decision to stand, hit, double-down, or spilt. Doubling-down and splitting are restricted to the house rules.

10, J, Q, and K are all counted as 10; Ace can be counted as 1 or 11; other cards are counted as their face points. When the total is 22 or over, it is considered as a busted hand. The dealer's hand needs to stand on 17 (all-17 or hard-17, depending on the house rules) and draw to 16. If the player busts, he/she automatically loses, no matter what score the dealer is. If the player does not bust, he/she automatically wins when the dealer busts. If both the dealer and the player do not bust, whoever has the higher score wins, and it is tie (push) when they have the same score.

If the original two cards are Ace and a 10-count card, it is a Natural Blackjack, which beats all non-natural blackjack hands. When the upcard is an Ace, players have the opportunity to buy insurnace. The insurance bet is a side bet in front of palyer's original bet. It's bet size is up to 1/2 of the original bet. Insurance bets win only when the dealer has a natural blackjack.

All winning bets are paid even money except the Natural Blackjack and the insurance bet. The natural blackjack gets a 3:2 payout (it may be changed depending on the house rules) and the insurance bet gets a 2:1 payout.

Splitting can only be played when the first two cards have the same point. Splitting hands need to have even money on each bet and they are played one by one as the regular hand. Splitting Ace's only gets one card on each hand and resplitting may not be allowed.

Doubling-down can only be played when the play has exact 2 cards. The player needs to increase the bet size to up to 2 times of the original bets and only one card is dealt to the doubling-down hand. Depending on the house rules, doubling-down may be restricted to the total of 10 or 11, and may not be allowed on splitted hands.

Hand Signals

Although you can communicate with the dealer, only your recogizable hand signals will be taken. All players, one by one, will be asked to use the standard hand signals to play their hands. (The game cannot be dealt backward without pit boss' permission.) We describe these standard hand signals as below.

In a face-up game:

Hit	Scratch your hand on the table from the front to back.
Stand	Flat your palm and wave your hand horizontally.
Double Down	Put a second bet next to your original bet (do not touch your original bet at all times during the game) and tell the dealer that you want to "double-down."
Split	Put a second bet next to your original bet and tell the dealer that you want to "split." Usually there is no confussion between doubling-down and splitting unless those two cards are a pair.

In a face-down game:

Hit	Hold your cards and scratch them on the table from the front to back.
Stand	Slide cards into your bet and leave them there. Do not use your hand to touch your bet. It is easier to use corners of your cards to slide cards into your bet. If necessary, flat your palm and wave your hand horizontally after you slide cards into your bet.
Double Down	Flip your two cards in front of your bet and put a second bet next to your original bet. If necessary, tell the dealer that you want to "double-down."
Split	Flip your two cards in front of your bet, separate them (left-right), and put a second bet next to your original bet. If necessary tell the dealer that you want to "split." After you split your cards, you then play these hands, one by one, in the same way as the face-up game.

Notations

The following notations are frequently used in blackjack books.

DOA	doubling-down on any first two cards is allowed
D10	doubling-down is allowed only when the first two cards sum to a total of 10 or 11
S17	the dealer stands on soft 17 (a soft 17 hand contains an Ace and can be treated as a total of 7 or 17)
H17	the dealer hits on soft 17
DAS	doubling-down-after-splitting is allowed
RSA	re-splitting-aces is allowed
LSR	late-surrender is allowed

Tipical House Rules

Although we specify the tipical rules here, casinos may have different rules.

Strip:
DOA, S17, DAS, no LSR, natural blackjack paid 3:2, one card for Ace-splitting and no replitting Ace's, and replitting up to 4 hands

Downtown Las Vegas:
DOA, H17, RSA, no DAS, no LSR, natural blackjack paid 3:2, one card for Ace-splitting, and replitting up to 4 hands

Atlantic City:
DOA, S17, DAS, natural blackjack paid 3:2, one card for Ace-splitting and no replitting Ace's, and no replitting

Reno:
D10, H17, no DAS, natural blackjack paid 3:2, one card for Ace-splitting and no replitting Ace's, and replitting up to 4 hands

The Basic Strategy

Assume the upcard is X and the player's hand is Y+Z. If we compute the expected gain/loss of all different actions of this situation, we can find out the correct action which gives the player the maximum gain (or the minimum loss). So, after computering the following table and choosing the best action, we can create the Table of the Basic Strategy.

   X  Y  Z | Stand | Stand on 12 | Stand on 17 | Double down | Split
  ---------|-------|-------------|-------------|-------------|-------
           |       |             |             |             |

Table of the Basic Strategy

The following table shows the Basic Strategy with the following conditions:
DOA (if D10, ignore unsuitable entries), S17 or H17, DAS or no DAS, no LSR (see Surrender section for more information), 1-, 2-, 4-, 6-, or 8-deck

  Player's   |<--------------------- Upcard ----------------->|
    Hand       2    3    4    5    6    7    8    9   10    A

    A-A       spl  spl  spl  spl  spl  spl  spl  spl  spl  spl
   10-10      ---  ---  ---  ---  ---  ---  ---  ---  ---  ---
    9-9       spl  spl  spl  spl  spl  ---  spl  spl  ---  ---
    8-8       spl  spl  spl  spl  spl  spl  spl  spl  spl  spl
    7-7       spl  spl  spl  spl  spl  spl   *   ---  ---  ---
    6-6       spl  spl  spl  spl  spl   *   ---  ---  ---  ---
    5-5       ---  ---  ---  ---  ---  ---  ---  ---  ---  ---
    4-4       ---  ---   *    *    *   ---  ---  ---  ---  ---
    3-3        *    *   spl  spl  spl  spl   *   ---  ---  ---
    2-2        *   spl  spl  spl  spl  spl  ---  ---  ---  ---

    11        dbl  dbl  dbl  dbl  dbl  dbl  dbl  dbl  dbl  dbl
    10        dbl  dbl  dbl  dbl  dbl  dbl  dbl  dbl  ---  ---
     9        dbl  dbl  dbl  dbl  dbl  ---  ---  ---  ---  ---
     8        ---  ---  ---  dbl  dbl  ---  ---  ---  ---  ---
    A-8       ---  ---  ---  ---  dbl  ---  ---  ---  ---  ---
    A-7       ---  dbl  dbl  dbl  dbl  ---  ---  ---  ---  ---
    A-6       dbl  dbl  dbl  dbl  dbl  ---  ---  ---  ---  ---
    A-5       ---  ---  dbl  dbl  dbl  ---  ---  ---  ---  ---
    A-4       ---  ---  dbl  dbl  dbl  ---  ---  ---  ---  ---
    A-3       ---  ---  dbl  dbl  dbl  ---  ---  ---  ---  ---
    A-2       ---  ---  dbl  dbl  dbl  ---  ---  ---  ---  ---

  Soft-19+    std  std  std  std  std  std  std  std  std  std
  Soft-18     std  std  std  std  std  std  std  ---  ---  std
  Soft-17-    ---  ---  ---  ---  ---  ---  ---  ---  ---  ---
  Hard-17+    std  std  std  std  std  std  std  std  std  std
  Hard-16     std  std  std  std  std  ---  ---  ---  ---  ---
  Hard-15     std  std  std  std  std  ---  ---  ---  ---  ---
  Hard-14     std  std  std  std  std  ---  ---  ---   #   ---
  Hard-13     std  std  std  std  std  ---  ---  ---  ---  ---
  Hard-12     ---  ---  std  std  std  ---  ---  ---  ---  ---

  Note: * --- split if DAS
        # --- stand on 7-7 against the dealer's 10
              in a single-deck game
        Do not take insurance

Surrender

There are two kinds of surrenders: Early (ESR) and Late (LSR). The early surrender allows players to surrender their hands before the upcard is flipped; the late surrender allows players to surrender their hands after the upcard is flipped and before they play their hands. Note that surrender cannot be applied to the dealer's nature blackjack hand. Rarely a casino offers ESR, because it is too favarable to players. When a player surrenders his/her hand, 1/2 of his/her bet will be returned.

The Basic Strategy for LSR is listed below.

Single-deck

Counting Systems

It had been shown that when 10-count cards and Ace cards are rich in the rest of the decks, the situation is more favorable to the Basic Strategy players. On the other hand, when low cards are rich, it is less favorable to players. Based on this conclusion, several counting systems were proposed. In this section, we only address 3 of them: Hi-Lo, Hi-Opt I, and 10's Count.

All of these counting systems assign different weights on cards and use the true count to determine the betting and playing strategies. The true count is equal to the running count divided by the number of unused decks. That is, when there are 1 1/2 decks left and the running count is +3, the true count is +3/1.5 = +2.

Weights of different cards: (N decks)

              2  3  4  5  6  7  8  9 10  A  Starting
 ---------------------------------------------------
 Hi-Lo       +1 +1 +1 +1 +1  0  0  0 -1 -1     0
 Hi-Opt I     0 +1 +1 +1 +1  0  0  0 -1  0     0
 10's Count  +1 +1 +1 +1 +1 +1 +1 +1 -2 +1   -4*N
 ---------------------------------------------------

The law does allow all casinos to "bar" players, especially card counters. If a player is barred and tries to re-enter the game, he/she will be arrested. To avoid being barred, players should not spread their bets too much. A 1-4 spread is usually recommended in blackjack books.

Simulation Results

Based on the Strip rules and the Basic Strategy, we simulate a single-deck game for more than 10,000,000 hands and the result shows that the final gain/loss is about -0.2% to +0.2%.

The following data show the winning percentage with differnt spread by using the Hi-Lo counting system along with the Basic Strategy and proper insurrance bets.

Condition: Single deck, 5 players, 2 hands per deck before shuffling, $2 bet for the first hand and for the cases that the running count <= 0, ($2 ?Spread) bet for the cases that the running count >= 1.

Total number of rounds = 3 million hands

Spread	Total Gain	Total Bet Size	Winning Percentage
1 - 1 (Even Bet)	$470	$67,800,813	+0.000693%
1 - 2	$324,040	$82,080,155	+0.394785%
1 - 3	$592,103	$96,330,398	+0.614659%
1 - 4	$912,261	$110,603,098	+0.824806%
1 - 5	$1,188,512	$124,946,639	+0.951216%
1 - 6	$1,466,368	$139,128,382	+1.053968%
1 - 7	$1,681,093	$153,518,640	+1.095042%
1 - 8	$1,985,563	$167,582,357	+1.184828%