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A game of Blackjack with a blackjack hand of 21.

Blackjack, also known as Twenty-one, Vingt-et-un (French for Twenty-one), or Pontoon, is the most widely played casino banking game in the world. The standard game is played with one or more Anglo-American decks containing 52 cards. The basic rules of the game involve adding the value of an initial two card hand in hopes of being dealt a value of twenty-one. If a value of less than twenty-one is dealt, the player may choose to be dealt single cards until they either reach a value of twenty one, reach a value they feel comfortable to play, or reach a value that exceeds twenty-one. The winner holds a hand with a value of, or nearest to, twenty-one without exceeding it. The game is played in many variations at casinos with different table rules. Much of blackjack's popularity is due to the mix of chance, skill, and the publicity that surrounds card counting (calculating the probability of advantages based on the ratio of high cards to low cards). The casino version of the game should not be confused with the British card game Black Jack (a variant of Crazy Eights).


Blackjack's precursor was "twenty-one," a game of unknown origin. The first written reference is found in a book of Miguel de Cervantes, the author of Don Quixote, and a gambler himself. The main characters of his tale "Rinconete y Cortadillo", from "Novelas Ejemplares", are a couple of cheaters working in Sevillemarker. They are proficient at cheating at "veintiuna" (Spanish for twenty-one), and stated that the object of the game is to reach 21 points without busting, and that the Ace values 1 or 11. The game is played with Baraja, that is without tens, which makes the game similar to the current Spanish 21. This short story was written between 1601 and 1602, so the game was played in Castilia since the beginning of the 17th Century or even earlier. Later references of this game are found in France and Spain.

When 21 was introduced in the United States it was not very popular, so gambling houses tried offering various bonus payouts to get the players to the tables. One such bonus was a 10-to-1 payout if the player's hand consisted of the ace of spades and a black Jack (either the Jack of clubs or the Jack of spades). This hand was called a "blackjack" and the name stuck to the game, even though the bonus payout was soon abolished. In the modern game, a "natural" or "blackjack" is simply an ace plus a ten-value card.

Rules of play against a casino

In casino blackjack, the dealer faces one to seven players from behind a kidney-shaped table. Each player plays his hand independently against the dealer. At the beginning of each round, the player places a bet in the "betting box" and receives an initial hand of two cards. The object of the game is to get a higher card total than the dealer, but without going over 21 which is called "busting", "breaking", or many other terms. Cards with a number 2 to 10 printed on them count as that value; the jack, queen, and king (also known as "face cards") count as 10; and the ace card can be either 1 or 11 at the player's choice. The player goes first and plays his hand by taking additional cards if he desires. If he goes over 21 points, he "busts" and automatically loses the hand and his bet. Then the dealer plays his or her hand. If the dealer busts, he loses to all remaining players who have cards whose values are equal to or below 21. If neither busts, the higher hand total wins. If a player ties with the dealer the hand is a "push" and the player's bet is returned. It is possible for the dealer to lose to some players but still beat other players in the same round.
Cards are dealt in three ways, either from one or two hand-held decks, from a box (known as a "shoe") containing four to eight decks, or from a shuffling machine. When dealt by hand, the player's two initial cards are usually face-down, while the dealer has one face-up card called the "upcard" and one face-down card called the "hole card." (In European blackjack, the dealer's hole card is not actually dealt until the players all play their hands.) When dealt from a shoe, all player cards are normally dealt face-up, with minor exceptions. It shouldn't matter to the non-expert player whether his cards are dealt face-down or face-up since the dealer must play according to predetermined rules. If the dealer has less than 17, he must hit. If the dealer has 17 or more, he must stand (take no more cards), unless it is a "soft 17" (a hand that includes an ace valued as "11," for example a hand consisting of Ace+6, or Ace+2+4). With a soft 17, the dealer follows the casino rules printed on the blackjack table, either to "hit soft 17" or to "stand on all 17's."

Normally, the highest possible hand is a "blackjack" or "natural," meaning an initial two-card total of 21 (an ace and a ten-value card). A player blackjack is an automatic winner unless the dealer also has blackjack, in which case the hand is a "push" (a tie). When the dealer upcard is an ace, the player is allowed to make a side bet called "insurance," supposedly to guard against the risk that the dealer has a blackjack (i.e., a ten-value card as his hole card). The insurance bet pays 2-to-1 if the dealer has a blackjack. Whenever the dealer has a blackjack, he wins against all player hands except those that also have a blackjack (which are a "push").

The minimum and maximum bets are posted on the table. The payoff on most bets is 1:1, meaning that the player wins the same amount as he bets. The payoff for a player blackjack is 3:2, meaning that the casino pays $3 for each $2 originally bet. (There are many single-deck games which pay only 6:5 for a blackjack. )

Player decisions

After receiving his initial two cards, the player has four standard options: he can "Hit," "Stand," "Double Down," or "Split a pair." Each option requires the use of a hand signal. At some casinos or tables, the player may have a fifth option called "Surrender."
  • Hit: Take another card from the dealer.
signal: (handheld) Scrape cards against table. (face up) Touch finger to table or wave hand toward himself.
  • Stand: Take no more cards; also known as "stick" or "stay".
signal: (handheld) Slide cards under bet. (face up) Wave hand horizontally.
  • Double down: After receiving his first two cards and before any more are dealt to him, a player has the option to "double down." This means the player is allowed to double his initial bet in exchange for receiving only one more card from the dealer. The hand played consists of his original two cards plus one more from the dealer. To do this he moves a second bet equal to the first into the betting box next to his original bet. (If desired and allowed by casino rules, the player is usually allowed to "double down for less," placing an amount less than the original bet next to it in the betting box, although this is generally not a good idea as the player should only double in favorable situations but should then increase the bet as much as possible. Adversely, a player cannot "double down for more" than the value of the original bet.)
signal: Place additional chips next to (not on top of) the original bet.
  • Split a pair: If his first two cards are a "pair," meaning two cards of the same value, the player can "split the pair." To do this, he moves a second bet equal to the first into an area outside the betting box of the original bet. The dealer separates the cards to create two hands, placing one bet with each hand. The player then plays two separate hands.
signal: Place additional chips next to the original bet outside of the betting box.
  • Surrender: Some casinos offer a fifth option called "surrender." After the dealer has checked for blackjack, the player may "surrender" by giving up half his bet and not playing out the hand.
signal: There is no commonly accepted hand signal; it is just done verbally.

Hand signals are used to assist the "eye in the sky," a person or video camera located above the table but concealed behind one-way glass. This tool is used to protect the casino against dealers or players who cheat. It may also be used to protect the casino against card counters, even though card counting is not illegal.

The player can take as many hits as he wants as long as the total in his hand is not above hard-20. However, if he busts, he loses that hand. After all players have finished making their decisions, the dealer then reveals his hole card and plays out his or her hand according to predetermined rules.

Rule variations and the "house advantage"

The blackjack player will encounter many rule variations which affect the house advantage and therefore affect his chances of winning. Some rules are determined by law or regulation, others by the casino itself. Not all rules are posted, so the player may have to ask either beforehand or when the situation occurs. Over 100 variations exist.

The casino has a "house advantage" at blackjack just as it does at any other casino game. The primary house advantage in blackjack comes from the fact that if the player busts he loses, irrespective of whether the dealer subsequently busts. If a particular casino game has a house advantage of 5%, it means that - over the long run - the casino will win about 5% of any initial bet. As long as the blackjack player uses the best possible strategy (a strategy which is known as "basic strategy"), the house advantage in blackjack is usually less than 1%. This is very favorable to the player compared to other casino games. Of course, many blackjack players do not know basic strategy or do not follow it, so the true house advantage in those cases may be much higher.

Dealer hits soft 17

Each casino has a rule about whether or not the dealer hits soft 17, a rule which is printed on the table itself. In the "S17" game, the dealer stands on all 17s. In the "H17" game, the dealer hits on soft 17s. Of course, the dealer always stands on hard 17s. In either case, the dealer has no choice; he either must or must not hit. The "Hit soft 17" game is less favorable to the player with about a 0.2% higher house advantage.

Number of decks

The number of decks used has a major effect on the player's chance of winning, because it affects the house advantage. All things being equal, fewer decks are always more favorable for the basic strategy player. One cause of this is that player blackjack is slightly more likely in single deck blackjack (because blackjack requires two different cards, by removing a card of one type (e.g., a Ten), getting one of a different type (e.g., an Ace) is more likely - and the effect is much greater in a single deck game than in a multi-deck game), and if the player does have blackjack, the dealer is significantly less likely to have blackjack as well (which is a push), meaning that statistically the player should get paid at 3:2 more often in the single deck game.

In reality, multi-deck games almost always have otherwise better rules than single-deck games. For illustrative purposes, the statistics below all use the same rules: double after split, resplit to four hands, one card to split Aces, no surrender, double on any two cards, original bets only lost on dealer blackjack, dealer hits soft 17, and cut-card used. The single deck game is much better than double deck, which is significantly better than four decks, while from six decks and up there is very little difference.

Number of Decks House Advantage
Single deck 0.17%
Double deck 0.46%
Four decks 0.60%
Six decks 0.64%
Eight decks 0.66%


Some casinos offer a favorable option called "surrender," which allows the player to give up half his bet and not play out the hand. This option is sometimes referred to as "late" surrender because it occurs after the dealer has checked his or her hole card for a blackjack. When casinos first opened in Atlantic City, the surrender option was available before the dealer checked for blackjack - a rule highly advantageous to the player - but this "early surrender" option soon disappeared. Early surrender variations still exist in several countries.

The player should only surrender on the very worst hands, because having even a 25% chance of winning will result in a better average return than giving up half of his bet. With early surrender, a player is more likely to surrender against a dealer Ace.


If the player splits a pair other than aces and a third card of that value appears, the player can usually split again (or "resplit") by putting up another bet equal to the original bet. Then there will be three bets on the table and three separate hands. Some casinos allow unlimited resplitting of cards other than aces, while others may limit it to a certain number of hands, such as four hands (for example, "resplit to 4").

Hit/resplit split aces

After splitting aces, one common rule is that only one card will be dealt to each ace; the player cannot split, double, or take another hit on either hand. Rule variants include allowing resplitting aces or allowing the player to hit split aces. Allowing the player to hit aces reduces the casino edge by about 0.13%, allowing resplitting aces reduces the edge by about 0.03%.

Double after split

After splitting a pair, some casinos allow the player to "double down" on either or both of the new two-card hands. This is called "double after split" and provides an advantage to the player of about 0.12%.

Double on 9/10/11 or 10/11 only

Often called "Reno" rules, this rule restricts the player to doubling down only on an initial player total of 10 or 11 (sometimes 9, 10, or 11 - more common in Europe). It prevents doubling on soft hands such as soft 17 (ace-6), and is unfavorable for the player. It increases the house advantage by between 0.09% (8 decks) and 0.15% (1 deck) for the 9-11 rule, and between 0.17% (8 decks) and 0.26% (single deck) for the 10-11 rule. These numbers can vary due to interaction with other rules.

European no-hole-card rule

In most non-U.S. casinos, a 'no hole card' game is played. This means that there is no dealer hole card. This usually affects the player's strategy when deciding whether to double and/or split since a dealer blackjack will result in the loss of the split and double bets. For instance, holding 11 against a dealer 10, the correct strategy is to double in a hole card game (where the player knows the dealer's second card is not an ace), but to hit in a no hole card game. The no hole card rule adds approximately 0.11% to the house edge.

In some places, if the dealer is later found to have blackjack, the player loses only his original bet but not any additional bets (doubles or splits). This has the same advantage as the usual game, and as such does not change basic strategy.

Altered payout for blackjack

In some places, a blackjack hand pays only 6:5 or even 1:1 instead of the usual 3:2. This is the most unfavorable common variation, increasing the house edge significantly more than most U.S. player restrictions. Since blackjack occurs in approximately 4.8% of hands, the 1:1 game increases the house edge by 2.3%, while the 6:5 game adds 1.4% to the house edge. The 1:1 payout for video blackjack is a key reason why it has never approached the table version in terms of popularity. The 6:5 rule is most commonly employed on table blackjack at single deck games - which are otherwise the most attractive game for a basic strategy player.

Dealer wins ties

Allowing the dealer to win all push hands is catastrophic to the player. Though rarely used in standard blackjack, it is sometimes seen in "blackjack-like" games.


If the dealer's upcard is an ace, the player is offered the option of taking insurance before the dealer checks his or her 'hole card'.

Insurance is a side bet of up to half the original bet placed on a special portion of the table usually marked "Insurance Pays 2 to 1". This side bet is offered only when the dealer's exposed card is an ace. The idea is that the dealer's second card has a fairly high probability (nearly one-third) to be ten-valued, giving the dealer a blackjack and almost results in a certain loss for the player. It is attractive (although not necessarily wise) for the player to insure against this possibility by making an "insurance" bet, which pays 2-to-1 if the dealer has a blackjack, in which case the "insurance proceeds" will make up for the concomitant loss on the original bet. The insurance bet is lost if the dealer does not have blackjack, although the player can still win or lose on the original bet.

Insurance is a poor bet for the player unless he is counting cards because, in an infinite deck, 4/13 of the cards have a value of ten (10, J, Q, or K) and 9/13 therefore are not, so the theoretical return for an infinite deck game is 4/13 * 2 * bet - 9/13 * bet = -1 /13 * bet, or -7.69%. In practice, the average house edge will be lower than this, because by eliminating even one non-ten card from the shoe (the dealer's ace), the proportion of the remaining cards that are valued at ten is higher. Even so, the bet is generally to be avoided, as the house's average edge is still more than 7%.

A player who is counting cards can keep count of the remaining tens in the shoe and use it to make insurance bets only when he has an edge (e.g., when more than one third of the remaining cards are tens). In addition, in a multi-hand single deck game, it is possible for insurance to be a good bet simply by observing the other cards on the table - for an initial hand, if the dealer has an ace, then there are 51 cards left in the deck, of which 16 are tens. However, if there are as few as 2 players playing, and none of their two initial cards are tens, then that means that 16 out of 47 remaining cards are ten - better than 1 in 3, and so the insurance bet is a good one.

When the player has blackjack and the dealer has an ace, the insurance bet may be offered as "even money", meaning that the player's blackjack is paid immediately at 1:1 before checking the dealer's hand. 'Even money' is just a way of expressing the actual payout on the insurance bet, given that the player has blackjack, it is not a different bet, and taking even money is generally even worse than average, because one of the player's two cards is tens, so the proportion of tens remaining in the deck is lower.

In casinos where a hole card is dealt, a dealer who is showing a card with a value of ace or 10 may slide the corner of his hole card over a small mirror or electronic sensor on the tabletop in order to check whether he has a blackjack. This practice minimizes the risk of inadvertently revealing the hole card, which may give the sharp-eyed player a considerable advantage.

Side bets

Some casinos offer a side bet with their blackjack games. Examples include side bets based on getting three 7s, a three card poker-style bet, a pair, and many others. For the side bet, the player will typically put up an additional wager alongside his main bet and can win or lose the side bet, regardless of the main game result. As there is little or no strategy involved, the house edge for side bets is usually much higher than the main game.

Blackjack strategy

Basic strategy

Because blackjack has an element of player choice, players can reduce casino advantage by playing optimally. The complete set of optimal plays is known as basic strategy. There are slight variations depending on the house rules and number of decks.

Your hand Dealer's face-up card
2 3 4 5 6 7 8 9 10 A
Hard totals (excluding pairs)
17-20 S S S S S S S S S S
15 S S S S S H H H SU H
13-14 S S S S S H H H H H
12 H H S S S H H H H H
11 Dh Dh Dh Dh Dh Dh Dh Dh Dh H
10 Dh Dh Dh Dh Dh Dh Dh Dh H H
9 H Dh Dh Dh Dh H H H H H
5-8 H H H H H H H H H H
Soft totals
2 3 4 5 6 7 8 9 10 A
A,8 A,9 S S S S S S S S S S
A,7 S Ds Ds Ds Ds S S H H H
A,6 H Dh Dh Dh Dh H H H H H
A,4 A,5 H H Dh Dh Dh H H H H H
A,2 A,3 H H H Dh Dh H H H H H
2 3 4 5 6 7 8 9 10 A
10,10 S S S S S S S S S S
5,5 Dh Dh Dh Dh Dh Dh Dh Dh H H
4,4 H H H SP SP H H H H H
2,2 3,3 SP SP SP SP SP SP H H H H
The above is a basic strategy table for 3 or more decks, dealer stands on soft 17, double on any 2 cards, double after split allowed, dealer peeks for blackjack, and blackjack pays 3:2.Key:
S = Stand
H = Hit
Dh = Double (if not allowed, then hit)
Ds = Double (if not allowed, then stand)
SP = Split
SU = Surrender (if not allowed, then hit)

Most Las Vegas Stripmarker casinos hit on soft 17. This rule change requires a slightly modified basic strategy table: double on 11 verses a dealer's upcard being an ace, double on A/7 verses a dealer 2, and double on A/8 verses a 6. Most casinos outside of Las Vegas still stand on soft 17.

Card counting

Basic strategy provides the player with the optimal play for any blackjack situation based on billions of hands played in the long run. By keeping track of the cards that have already been played, it is possible to know when the cards remaining in the deck are advantageous for the player.

Card counting creates two opportunities:
  • The player can make larger bets when he has the advantage. For example, the player can increase the starting bet if there are many aces and tens left in the deck, in the hope of hitting a blackjack.
  • The player can use information about the remaining cards to improve upon the basic strategy rules for specific hands played. For example, with many tens left in the deck, the player may double down in more situations since there is a better chance of making a strong hand.

Virtually all card-counting systems do not require the player to remember which cards have been played. Rather, a point system is established for the cards, and the player keeps track of a simple point count as the cards are played out from the dealer.

Depending on the particular blackjack rules in a given casino, basic strategy reduces the house advantage to less than 1%. Card-counting, if done correctly, can give the player an advantage, typically ranging from 0 to 2% over the house.

Card-counting mentally is legal and is not considered cheating. However, most casinos have the right to ban players, with or without cause, and card-counting is frequently used as a justification to ban a player. Usually, the casino will inform the player that he is no longer welcome to play blackjack at that casino and may be banned from the property. Players must be careful not to signal the fact that they are counting, and the use of electronic or other counting devices is usually illegal.
See also: MIT Blackjack Team

Composition-dependent strategy

Basic strategy is based on a player's point total and the dealer's visible card. A player's ideal decision may depend on the composition of his hand, not just the information considered in the basic strategy. For example, a player should ordinarily stand when holding 12 against a dealer 4. However, in a single deck game, the player should hit if his 12 consists of a 10 and a 2; this is because the player wants to receive any card other than a 10 if hitting, and the 10 in the player's hand is one less card available to cause a bust for the player or the dealer.

However, in situations where basic and composition-dependent strategy lead to different actions, the difference in expected value between the two decisions will be small. Additionally, as the number of decks used in a blackjack game rises, both the number of situations where composition determines the correct strategy and the house edge improvement from using a composition-dependent strategy will fall. Using a composition-dependent strategy only reduces house edge by 0.0031% in a six-deck game, less than one tenth the improvement in a single-deck game (0.0387%).

Shuffle tracking and other advantage-play techniques

Techniques other than card-counting can swing the advantage of casino blackjack towards the player. All such techniques are based on the value of the cards to the player and the casino, as originally conceived by Edward O. Thorp. One technique, mainly applicable in multi-deck games, involves tracking groups of cards (aka slugs, clumps, packs) during the play of the shoe, following them through the shuffle and then playing and betting accordingly when those cards come into play from the new shoe. This technique, which is admittedly much more difficult than straight card-counting and requires excellent eyesight and powers of visual estimation, has the additional benefit of fooling casino employees who are monitoring the player's actions and the count, since the shuffle tracker could be, at times, betting and/or playing opposite to how a straightforward card-counter would.

Arnold Snyder's articles in Blackjack Forum magazine brought shuffle tracking to the general public. His book, The Shuffle Tracker's Cookbook, mathematically analyzed the player edge available from shuffle tracking based on the actual size of the tracked slug. Jerry L. Patterson also developed and published a shuffle-tracking method for tracking favorable clumps of cards and cutting them into play and tracking unfavorable clumps of cards and cutting them out of play.
Other legal methods of gaining a player advantage at blackjack include a wide variety of techniques for hole carding or gaining information about the next card to be dealt. In addition, match-play coupons give the skillful basic-strategy blackjack player an edge. And finally, a special promotion - such as 2:1 for a blackjack - can temporarily swing the advantage to the player.


Pontoon is an English variation of blackjack with significant rule and strategy differences. However, in Australia and Malaysia, Pontoon is an unlicensed version of the American game Spanish 21 played without a hole card; despite the name, it bears no relation to English Pontoon.

Spanish 21 provides players with many liberal blackjack rules, such as doubling down any number of cards (with the option to 'rescue', or surrender only one wager to the house), payout bonuses for five or more card 21s, 6-7-8 21s, 7-7-7 21s, late surrender, and player blackjacks always winning and player 21s always winning, at the cost of having no 10 cards in the deck (though there are jacks, queens, and kings).

21st-Century Blackjack (also known as "Vegas Style" Blackjack) is commonly found in many California card rooms. In this form of the game, a player bust does not always result in an automatic loss; there are a handful of situations where the player can still push if the dealer busts as well, provided that the dealer busts with a higher total.

Certain rule changes are employed to create new variant games. These changes, while attracting the novice player, actually increase the house edge in these games. Double Exposure Blackjack is a variant in which the dealer's cards are both face-up. This game increases house edge by paying even money on blackjacks and players losing ties. Double Attack Blackjack has very liberal blackjack rules and the option of increasing one's wager after seeing the dealer's up card. This game is dealt from a Spanish shoe, and blackjacks only pay even money.

The French and German variant "Vingt-et-un" (Twenty-one) and "Siebzehn und Vier" (Seventeen and Four) don't include splitting. An ace can only count as eleven, but two aces count as a blackjack. This variant is seldom found in casinos but is more common in private circles and barracks.

Chinese Blackjack is played by many in Asia, having no splitting of cards, but with other card combination regulations.

Another variant is Blackjack Switch, a version in which a player is dealt two hands and is allowed to switch cards. For example, if the player is dealt 10-6 and 5-10, then the player can switch two cards to make hands of 10-10 and 6-5. Natural blackjacks are paid 1:1 instead of the standard 3:2, and a dealer 22 is a push.

In Multiple Action Blackjack the player places between 2 or 3 bets on a single hand. The dealer then gets a hand for each bet the player places on a hand. This essentially doubles the number of hands a single dealer can play per hour. Splitting and Doubling are still allowed.

Recently, thanks to the popularity of poker, Elimination Blackjack has gained a following. Elimination Blackjack is a tournament format of blackjack.

Many casinos offer optional side bets at standard blackjack tables. For example, one common side-bet is "Royal Match", in which the player is paid if his first two cards are in the same suit, and receives a higher payout if they are a suited queen and king (and a jackpot payout if both the player and the dealer have a suited queen-king hand). Another increasingly common variant is "21+3," in which the player's two cards and the dealer's up card form a three-card poker hand; players are paid 9 to 1 on a straight, flush or three of a kind. These side bets invariably offer worse odds than well-played blackjack.

In April 2007, a new version of blackjack, called "three card blackjack" was approved for play in the State of Washington and is played with one deck of 52 cards. In this version of the game, the players place an ante bet. The players and dealer are then dealt 3 cards each. The players make the best blackjack (21) hand they can using 2 or all 3 cards. If the player likes his hand he makes a play bet that is equivalent to the ante bet. The dealer must qualify with an 18 or better. If the dealer qualifies and the player beats the dealer, the player is paid 1-1 on both the Ante and Play bets. If the dealer does not qualify, the player is paid 1-1 on his Ante bet and Play bet pushes. There is no hitting and no busting. At the same time that the player makes the Ante bet, he has the option of making an "ace plus" bet. If the player has one ace in his hand of 3 cards, he gets paid 1-1. An ace and a 10 or face card pays 3-1. An ace and two 10's or face cards is paid 5-1. Two aces pays 15-1. Three aces pays 100-1.

Blackjack Hall of Fame

In 2002, professional gamblers around the world were invited to nominate great blackjack players for admission into the Blackjack Hall of Fame. Seven members were inducted in 2002, with new people inducted every year after. The Hall of Fame is at the Barona Casino in San Diegomarker. Members include Edward O. Thorp, author of the 1960s book Beat the Dealer which proved that the game could be beaten with a combination of basic strategy and card counting; Ken Uston, who popularized the concept of team play; Arnold Snyder, author and editor of the Blackjack Forum trade journal; Stanford Wong, author and popularizer of the "Wonging" technique of only playing at a positive count, and several others.


  1. Scarne's New Complete Guide to Gambling, p. 342
  2. 100+ Blackjack variations
  3. Blackjack Insurance Exceptions
  4. Blackjack side bets -- analyzed by The Wizard of Odds
  5. Rules & House Edge Table
  6. Theory of Blackjack, p. 5
  7. Theory of Blackjack, pp 6–7
  8. The Mathematics of Gambling
  9. Shuffle Tracking Counts
  10. The Gambling Times Guide to Blackjack; Gambling Times Incorporated, Hollywood, CA; © 1984; Page 110; ISBN 0-89746-015-4 Shuffle-Tracking An Easy Way to Start ]
  11. Break the Dealer; by Jerry L. Patterson and Eddie Olsen; Perigee Books; A Division of Penguin Putnam; © 1986; ISBN 0-399-51233-0 Shuffle-Tracking; Chapter 6, Page 83]
  12. Blackjack: A Winner’s Handbook; by Jerry L. Patterson; Perigee Books; A Division of Penguin Putnam; © 1990; ISBN 0-399-51598-4 Shuffle-Tracking; Chapter 4, Page 51]


  • Beat the Dealer : A Winning Strategy for the Game of Twenty-One, Edward O. Thorp, 1966, ISBN 978-0-394-70310-7
  • Blackbelt in Blackjack, Arnold Snyder, 1998 (1980), ISBN 978-0-910575-05-8
  • Blackjack: A Winner’s Handbook, Jerry L. Patterson, 2001, (1978), ISBN 978-0-399-52683-1
  • Ken Uston on Blackjack, Ken Uston, 1986, ISBN 978-0-8184-0411-5
  • Knock-Out Blackjack, Olaf Vancura and Ken Fuchs, 1998, ISBN 978-0-929712-31-4
  • Luck, Logic, and White Lies: The Mathematics of Games, Jörg Bewersdorff, 2004, ISBN 978-1-56881-210-6, 121-134
  • Million Dollar Blackjack, Ken Uston, 1994 (1981), ISBN 978-0-89746-068-2
  • Playing Blackjack as a Business, Lawrence Revere, 1998 (1971), ISBN 978-0-8184-0064-3
  • Professional Blackjack, Stanford Wong, 1994 (1975), ISBN 978-0-935926-21-7
  • The Theory of Blackjack, Peter Griffin, 1996 (1979), ISBN 978-0-929712-12-3
  • The Theory of Gambling and Statistical Logic, Richard A. Epstein, 1977, ISBN 978-0-12-240761-1, 215-251
  • The World's Greatest Blackjack Book, Lance Humble and Carl Cooper, 1980, ISBN 978-0385153829
Regulation in the United Kingdom

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