Acknowledgements
We would like to thank Professor Curtis Gary Dean for his input and
encouragement on this project. Without his expertise and ideas we would have had a
" completing the project. We would also like to thank Ken Smith,
difficult time
Web master of Blackjacklnfo.com, for his response to our questions. The statistics on his
website provided us with something credible to compare our results. His input showed us
that our program was similar to his Basic Strategy Engine, and that many others in the
blackjack community have verified his results. We would also like to thank Melissa
Cordial for her help in proofreading our paper. Without her help, we would have had a
difficult time in making the paper organized and readable.
2
Abstract
This paper will discuss the important elements of blackjack and give insight as to
the best strategy for bets and game decisions that one can play. There are many strategies
that people can find in literature, online, and on television. The goal of this paper is to
show that the optimal strategy combines the use of Basic Strategy, first developed by
Edward Thorpe, and card counting. Many other strategies might seem to be foolproof,
but, in the long run, most of these strategies will only make a player lose more money.
A computer simulation was written to simulate a real life game for a Basic
Strategy player, a hi-lo card counter, and a player who uses the Martingale betting
strategy. The Martingale strategy is just one of the many betting strategies that seem to
-
be better than the Basic Strategy, but only make a player worse off in the long run. The
program was set up to run large numbers of trials to get a good estimate of the long-term
advantage (or disadvantage) of a player in a game. The results of this program along
with statistics found through research are shown and analyzed in this paper.
3
Table of Contents
I. Introduction ........................................................................................................ 5
II. Casinos ............................................................................................................. 6
III. Mathematical Principles .................................................................................. 8
A. Basic Probabilities in Blackjack .......................................................... 8
B. Basic Strategy .................................................................................... 11
C. Gambler's Ruin ................................................................................. 13
IV. Betting Systems ............................................................................................ 16
A. Martingale Betting System ................................................................ 16
B. Kelly Betting Criterion ...................................................................... 17
C. Climbing Betting System .................................................................. 18
V. Card Counting ................................................................................................ 19
A. Credibility of Count .......................................................................... 19
B. Hi-Lo Count ...................................................................................... 20
C. Core Strategy ..................................................................................... 22
VI. Setup of the Program .................................................................................... 24
VII. Analysis
A.
B.
C.
D.
.................................................................................................... 26
Multiple Deck Games ....................................................................... 26
Common Human Errors .................................................................... 29
Results ............................................................................................... 31
Online Blackjack ............................................................................... 36
VIII. Conclusion .................................................................................................. 38
IX. Strategy Charts .............................................................................................. 40
A. Basic Strategy Charts ........................................................................ 40
1. One Deck Charts ................................................................... 41
2. Two Deck Charts .................................................................. 43
3. Three Deck Charts ............................................................... .45
4. Four Deck Charts .................................................................. 47
5. Five Deck Charts ................................................................... 49
6. Six Deck Charts .................................................................... 51
7. Seven Deck Charts ................................................................ 53
8. Eight Deck Charts ................................................................. 55
B. Core Strategy Charts ......................................................................... 57
X. Works Cited ................................................................................................... 61
4
I. Introduction
The game of blackjack, or Twenty-one, is one of the world's most widely played
gambling games. The object is simple, draw cards and get as close to 21 as possible
without going over. But the game itself is much more complex. The dealer deals cards
out to the player, and to himself. When all the cards are dealt and the player's hand total
is higher than the dealers without going over, then he wins an amount equal to his total
bet. If the player gets dealt an Ace and a ten-valued card, while the dealer has less than
21, then the player gets a blackjack and he is paid one and half times his initial bet. If the
dealer has the higher hand total in the end, the player loses his wager. If the hand totals
for the player and the dealer are the same, then it is a push and no money exchanges
-
hands. If either a player or the dealer draws a card and their respective hand total is
greater than 21, this is known as a bust. If a player busts, he loses his bet on that
particular hand. If the dealer busts, any player on the table who has not busted wins their
respecti ve bet.
The dealer does not make any decisions. On any hand value of 16 or lower he
must take a card, and any value 17 or higher, he must stand (in most situations). The
player is more flexible as to what he can do. After a player's first two cards are dealt a
player can decide to double down, split, hit, or stand. When a player doubles down, he
doubles his initial bet and only gets one additional card. A player can split his first two
cards, if they are of equal value, and play two hands with each card as the first card dealt
in each hand and a bet equal to his initial bet on both hands. If a player decides to hit, he
decides to take another card (this can be done as much as the player likes until he decides
5
cards, if they are of equal value, and play two hands with each card as the first card dealt
in each hand and a bet equal to his initial bet on both hands. If a player decides to hit, he
decides to take another card (this can be done as much as the player likes until he decides
to stand, or once he busts). If a player decides to stand, the player waits to see what the
dealer has versus his present total.
All too often people play gambling games and only look at what they can win and
do not assess what they are expected to win/lose. Blackjack is one of the most popular
gambling games even though the player usually holds a disadvantage to the house. This
disadvantage, along with a house advantage in many other games, amounts to extremely
large gains for the casino. This disadvantage in blackjack, however, is directly related to
how the game is played. Understanding the mathematics and probabilities behind
-
blackjack is the only effective way to beat the house. With these principles, along with a
little bit of luck, a person's night at the casino can be very profitable.
II. Casinos
Owning a casino is one of the most profitable careers one can have. Millions of
dollars are gambled every night, and the casino has an edge in virtually every game that it
offers. A 5% edge for casinos is common. This means that for every dollar a gambler
spends, the casino will on average make five cents. The gambler can expect to come
home with about ninety-five cents for every dollar he bets. The casino will almost
always enjoy this advantage. However, blackjack is one of the few games where the
player has the opportunity to gain an advantage.
6
The casino is not oblivious to this fact. According to Blackjack for Winners by
Scott Frank, the success of players has drastically changed the game of blackjack over the
years. In the early 1960s, a mathematician named Edward Thorpe devised a method of
tracking lO-valued cards using a plus and minus system, generically referred to as card
counting. He combined it with a computer-devised strategy that identified the most
favorable response to any situation. It was so successful that it single-handedly caused
the casinos to change the rules. There would be no more splitting Aces, and doubling
down was limited to hard eleven. Splitting Aces gives the player something to be excited
about, and doubling down is often considered the real moneymaker in blackjack. These
rule changes were having an impact on the popularity of the game. But with tourism
declining and dealers upset over lost tips, they reverted back to original rules. The
-
casinos would combat card counting by discriminating against counters.
One example of this discrimination was the Atlantic City casinos barring certain
people they suspected of using these potentially profitable practices. One notorious
counter named Ken Uston successfully sued Atlantic City casinos for not allowing him or
his associates to play. The lawsuit forbade the casinos from barring or discriminating in
any way against any players who were simply using their intelligence to make decisions.
The major outcomes of this were some very diminutive games for the player. More
decks were used to take back some of the advantage that a player gains by using a card
counting strategy. Using more decks makes it more unlikely that a player will have a
count that is to their advantage. Eight-deck games did not exist prior to this court
decision. All Atlantic City games were four-decks and six-decks until that point.
7
"-
Although it is still possible to count cards in an eight-deck shoe in Atlantic City, most
people do not have the time nor the patience to learn.
Las Vegas casinos have not lost this right. Casinos there reserve the right to ban
anyone whom they do not want in their establishment, including those who are just lucky.
But most Vegas casinos will not do this because they understand that luck is part of the
game. The only people they want out of their establishments are those who cheat and
those whom they suspect are counting cards. Therefore, the counter must be consistent,
careful, and discreet in order to blend in.
The reason the casinos are so concerned with blackjack is that the decisions the
player makes have a much larger influence on the game as compared to most other
games. In most games, the bet is set at the beginning of play. In blackjack, a player can
put more money on the table when he knows that he has an advantage by doubling down
or splitting. Because of these decisions, a player can take a lot or all of the advantage
away from the house.
III. Mathematical Principles
III. A. Basic Probabilities in Blackjack
Without strategy, you could be giving up to a 40% edge in favor of the house
(Vogel). But by simply memorizing Basic Strategy, that number can be reduced to a
mere .5%, even for multiple deck games. By including card counting techniques and
betting strategies, your chances of winning are even better than that. To come up with
these strategies, you simply have to look at the game of blackjack as a mathematical
model.
8
In every situation of blackjack, mathematical principles can tell you whether or
not you are playing a favorable game. The chart below shows the probabilities of the
dealer busting based on his upcard. Ideally, a player will be dealt a hand he is
comfortable sitting on and the dealer will bust.
Dealer's Probability of Busting
Dealer Upcard
A
2
3
4
5
6
7
8
9
10
Chance of Bust
11.65%
35.30%
37.56%
40.28%
42.89%
42.08%
25.99%
23.86%
23.34%
21.43%
According to the chart above, the dealer will have a higher probability of busting when
his upcard is a 5 or 6. The dealers overall busting probability is 28.36%. This means the
player will have to rely more on his own good fortune, than the dealer's bad fortune.
Two other interesting charts to observe come from Blackjack For Winners. These
charts are shown below.
Player's Expectation of Winning
Hand Total
17 or less
18
19
20
21
Natural Blackjack
Probability
28.36%
42.94%
56.75%
70.23%
87.81%
95.17%
.9
Dealers Final Total Probabilities
Probability
14.58%
13.81%
13.48%
17.58%
7.36%
4.83%
Hand Total
17
18
19
20
21
Natural Blackjack
Bust
28.36%
The charts deal with the player's expectation of winning and the dealer's final total
probability. If a player stands on 17 or less, he will only win 28.36% of the time.
Standing on a hand this low means the player is counting on the dealer to bust. Since the
dealer is forced to hit on any hand below 17, the player can only win by the dealer
busting. An 18 is commonly considered a safe hand, but according to the chart the player
will only win 42.94% of the time. Statistically, the player will push 13.81 % of the time.
This means the hand is over, and no money has been exchanged. By adding the two
percentages up, it can be concluded that 56.75% of the time a player has an 18 he will not
lose money. If a particular player is satisfied by either winning or forcing a push, then a
19 will statistically give the player a 70.23% chance of being satisfied. Most blackjack
players do not play to tie, they play to win. The lowest hand total that will give the
player a statistically higher chance of winning than not winning is a 19.
A good strategy should take this into account but should not necessarily target
getting a 19. Certainly a player should not give up an 18 and risk busting because he is
not yet at 19 or above. A player's strategy needs to be flexible as to the situation which
he is in. Using statistics, one can optimize the delicate balance in the risk of taking
another card and obtaining a higher total and playing it safe without the chance of bust.
10
III. B. Basic Strategy
Basic strategy optimizes the balance between taking another card or standing. For
example, why would a player stand on a soft 18 (having an Ace on top of a total of seven)
when the dealer shows a two-eight? As mentioned above, the player's goal should be to
get a 19 to maximize his shot at winning. This is because there are more cards that could
hurt the player than help him given an even deck. Drawing a 10 will give the player the
same value. The additional 10 will make the player's hand a hard hand because the Ace
will now be valued as a one, but his new total will remain the same. Drawing a four
through a nine will actually hurt the player's hand. Drawing a two, three, or an Ace can
help the player. The decision to hit should be based on the dealer's assumed total, if the
dealer has a 10 showing, one would probably take another card but not put more money
on the table by doubling down. Whereas a six for the dealer's upcard would be a good
opportunity for the player to get more money on the table in order to improve his
expected winnings in that situation. This decision process that takes in to account how
many cards will help the player or hurt the player and whether or not to put more money
on the table is the basis for Basic Strategy.
As mentioned above, doubling down helps get more money on the table for
favorable situations for the player. A player may double the amount of his initial bet
after looking at his first two cards. Once this is done the player can receive only one
additional card. The double down option allows us to double our bets and our profits at
the best possible time, when we have the edge over the dealer on a hand. If the dealer
-
shows a two through a six, according to the above chart he will bust between 35 and 42
11
percent of the time. This would be an opportune time for the player to double down,
especially when he is showing a nine, 10 or 11. In this situation a player will have two
things going for him. First, the dealer is likely to bust, and second, if the dealer does not
bust the player is more likely to beat the dealer's hand because he would have a high
probability of a 19,20, or 21. Many casinos place restrictions on when a player may
double down. Some casinos only allow doubling down.on hand values of 10 or 11, some
do not allow doubling down on soft hands, and some do not allow doubling down after
splitting. The less restrictions placed on doubling down, the more advantageous it is for
the players.
•
Another important part of Basic Strategy in blackjack is splitting pairs. If a player
is dealt a pair of the same value, he has the option of splitting the pair into two separate
hands and make separate bets on each hand. For instance if a player gets dealt two Aces,
he will choose to split them and play them like he has two separate hands. The two Aces
means he has two chances of drawing a 10 for a 21 (this is no longer considered a
blackjack however). A player should choose not to split his lOs because he would
already have a 20, which is already a probable winner. Basic strategy dictates a player
should split his nines even though an 18 is a standing hand. As mentioned above, an 18
has a better chance of losing than winning. Splitting nines will give the player a better
shot at a 19, which will in turn give him a better chance of winning.
Basic strategy is not a guaranteed winner, however. A player will hit high streaks
and low streaks. To accommodate for these streaks, a player needs to have a large
bankroll to make it a significantly small probability that one of those low streaks will bust
him out.
12
III. C. Gambler's Ruin
Gambler's ruin calculates the probability of busting out before hitting a specific
income level. Certainly in a negative expectation game, the more you want to win, the
less likely it is that you get to that level. Also, it is less likely to get to a high level if you
do not have enough to ride out the losing streaks that are bound to happen. Knowing
when to stop is an important part of any player's strategy. Players often go into a casino
looking to double their money, but this "greed" often causes them to come out emptyhanded. Gambler's ruin looks at the chances of a player reaching a certain wealth before
going broke.
Gambler's ruin has been looked at extensively over the years. Mathematician
-
Henry Tamburin set up a website looking at the topic. For the majority of blackjack
games, the casino will have an edge of about 0.5% over the Basic Strategy player.
Sometimes the player can reduce the edge by playing in a casino with better rules or
playing with fewer decks of cards. But according to Tamburin .5% is among the most
common house advantages. Using the gambler's ruin equation, a player has about an
88% probability of ruin (losing it all) before doubling a 200-unit bankroll (the amount of
money a player brings to gamble). To say it another way, he has about a 12% probability
of winning another 200 units (double) and 88% probability of losing the initial 200 units.
If he continues to play, he will most likely lose the 200 units before he ever gets ahead
200 units.
-
If the player decreases the amount of his bankroll from 200 to 100 units or less,
his chances to double will increase.
13
,-
Bankroll (units)
Probability of Doubling vs Ruin
10 20 50 100 200
47% 45% 38% 27% 12%
It is clear from the information above that the smaller the bankroll, the better chance a
player has of doubling his money. This makes sense because the larger the bankroll, the
more time you will have to spend at the casino before doubling your money. This gives
the casino's edge a longer time to start taking its toll.
Some players may take a more conservative approach by setting a more modest
goal. Instead of trying to double it, a player may decide to quit at a lower total. As
mentioned above a player only has a 12% chance of doubling a 200-unit bankroll.
However, the lower a players win goal, the higher probability a person with this bankroll
,-
will succeed. The following table summarizes chances of success of player with a 200unit bankroll and a win goal of 200, 100, and 20 units.
Starting Bankroll of 200 units.
200 100 20
Win Goal
% Probability of Winning vs. Ruin 12% 36% 45%
The chances of winning a modest amount for a Basic Strategy player are a lot better than
the chances of winning a large sum. So as a Basic Strategy player, win goals should set
at approximately 20%-30% of the starting bankroll. The more time spent at the table, the
more the probability of success will decrease.
A blackjack card counter will obviously have a different betting strategy than a
Basic Strategy player and a better chance of winning. Tamburin assumed a counter using
a simple plus-minus system in multiple deck game would have about a 0.5% edge over
the casino. Below are the counter's chances of doubling different size bankrolls vs ruin.
14
Bankroll (units)
Probability of Doubling vs Ruin
10 20 50 100 200
52% 55% 62% 73% 88%
The card-counters chances of doubling his bankroll versus ruin increase as the bankroll
increases. This is the opposite effect of a Basic Strategy player. This is because the
longer a counter sits at the table, the more he can play to his advantage. The card-counter
has an 88% chance of doubling his $200 if he plays to his advantage. The normal player
had a 12% chance. The slight advantage will give the card-counter a positive
expectation.
Similar to the Basic Strategy player, when the card counter starts with a $200
bankroll and decides to quit with a smaller profit, his probability of success gradually
Increases.
Starting Bankroll of 200 units
Win Goal
200 100 20
% Probability of Winning vs Ruin 88% 91% 97%
Since this player now has an edge, he has a greater chance of winning smaller amounts of
money with a $200 bankroll.
Successful gambling is all based upon probability and statistics; winning is based
on trends and bankroll. Statistics tell you that something is likely to happen. Probability
tells you it must happen during a certain period of time. Trends work to balance these
two together and bankroll determines if the player will reach a favorable trend.
Bankroll is the amount of money a player brings to a particular session. A rule of
thumb is that the bankroll should be approximately 40 times the initial wager. If a player
does not have a good bankroll, he'll play scared. He'll lose his strategy, self-control and
his patience. This frequently happens because players often try to make up for their
15
-
losses in a much riskier way than which they had initially played, and this often leads to
them losing all of their money in doing so. To win at blackjack, a player has to stay
comfortable with his betting strategy. In order to ensure this he should play with money
he can afford to lose because nothing is guaranteed in this game.
Memorizing Basic Strategy is the first part of becoming a successful blackjack
player. Basic strategy is, given a fair and even deck, the move that makes the most
mathematical sense. The Basic Strategy tables are shown in the appendix.
IV. Betting Systems
IV. A. Martingale Betting System
While Basic Strategy is the ideal strategy to play for the risk averse, many people
will combine it with another type of strategy, such as a betting strategy. One of the more
popular betting strategies in the game is the Martingale strategy. This is a very high-risk
strategy where you control the stakes. The Martingale is a system that has a very high
risk of going broke. A player needs a huge bankroll to fund this strategy. How this
strategy works is if he loses, he would double his bet until he wins. The attraction of this
strategy is when the player wins, he gains back his losses plus his initial bet. Another
money-maker in this game, is that if a player loses many times in a row and hits a
blackjack, he gets the initial bet back plus half of the money on the table, and that can be
a significant amount of money after six or seven losses. The problem is that the stakes
grow at a very fast rate, and when the maximum bet is reached and the player loses again,
he will have lost a significant amount of money and his chance to win it back.
16
.-
Originally the martingale system was used on a red-black bet for roulette. The
odds of winning the game are slightly below 50% (18/38 to be precise). This means that
the risk is minimized, but also that the profit at end of each sequence will be only the
starting-stake. For example, lets say that a roulette player has an initial bet of $5. He
loses the first hand and his bet is now up to $10. He then loses his next five hands. His
wager is now $320. If he wins his next hand, he will have won $320. Not bad, but he
had already lost $315 so his winnings were only $5. The player will have put up $320 to
win back $5. But in order to even play, he would have needed a bankroll of at least $615.
If the table limit on this table is $500, then he is finished after six losses in a row. He
probably lost his bankroll and a lot of pride. Losing six times in a row is certainly not
uncommon and will happen eventually in the long run. This is the same problem that is
faced in blackjack.
We all know that many losing streaks in blackjack run longer than six, eight or
even 10 in a row. The martingale quickly runs into the table limits. A player who starts at
a $2 table with a $500 upper limit is finished after nine losses in a row and is down over
$1000. It would take up to another 500 winning hands to make up this loss. Basically,
one losing streak will put the player in a hole he will never be able to climb out of. His
day of gambling will be finished in a hurry. This strategy seems to involve too much risk
for a small reward, but it could have its uses if a person were to set modest limits, as we
will deal with later in this paper. Basic strategy still is the basis for this and many other
betting strategies.
17
,IV. B. Kelly Betting Criterion
Optimal betting strategies have been developed through mathematical utility
theory. Famous blackjack mathematician Dr. Edward Thorpe introduced the Kelly
criterion for blackjack in his book Beat the Dealer. A Kelly betting scheme is loosely
one in which the player attempts to bet a proportion of his bankroll equal to his
percentage expectation. For instance, a player with a 2% advantage would bet roughly
2% of his bankroll. Here is a simple example. Forget for the moment that the game is
blackjack, and ignore the intricacies of the game. Assume that two players, each with a
$1000 bankroll, are betting in a game where one player has an advantage of 1% over the
-.
house, and the other player, due to a different strategy, has a 2% advantage. If both of
these players placed equal sized bets, then the player with the 2% advantage would
expect to win twice as much money as the player with the 1% advantage. If both players
were using a Kelly-type betting scheme, however, the player with the 2% advantage
would expect to win 4 times the expectation of the player with the 1% advantage. With a
Kelly betting scheme, the player with the 1% advantage would bet 1% of his bankroll, or
$10. His expectation on this bet would be 1% of $10, or 10¢. The player with the 2%
advantage, however, would make a bet of $20 (2% of his bankroll). His expectation on
his bet would be 2% of $20, or 40¢.
IV. C. Climbing Betting System
Climbing is a simple betting strategy designed for gamblers with the appropriate
bankroll. As described by Philip Vogel in Blackjack: The Real Deal, the player simply
18
looks for a hot table with favorable trends. He then starts with a wager, usually the
minimum bet, and plays with that wager until he wins three consecutive hands. He then
bumps the wager up one unit. If he wins twice in a row with that, he bumps it up another
unit. If one loss occurs, the player then goes down to the initial wager and starts again.
This loss will then be paid for by wins previous to it.
All these strategies look to maximize a player's wealth through changing the size
of the bet. This idea is also the basis for card counting. When a player has a significant
advantage, card counting dictates that a player increases his bet proportionately to the
initial bet. This act gains back money and nets the player an advantage in the right
circumstances. When counting, the true count gives the greatest estimate of an advantage
as the game nears the end of the deck.
V. Card Counting
V. A. Credibility of Count
In most casino blackjack games the deck will not be dealt down to the last
remaining cards. It would give a huge advantage to the card counter and as mentioned
above, any casino will take measures to protect against a player gaining an advantage.
But knowing if a deck is rich in certain valued cards will help a player make his
decisions. Card counting is simply a tool by which a player can approximate his
advantage over the house. When the approximate advantage is low, the player bets low
and minimizes losses. When the approximate advantage is high, the player puts more
money on the table knowing he has a better chance to win. Card counting does not mean
19
--
that a player knows exactly what card is coming next. He simply knows what card is
likely to come next.
The premise of card counting is that a deck or shoe that is rich in face cards or
Aces favors the player over the dealer. This is true for several reasons. A player who
receives a natural blackjack is paid 3:2 on his original bet, whereas the dealer only wins
the wager if he has blackjack. Also, if a dealer receives a poor hand on the first two cards
(a 12-16), they must draw another card. Since the deck is rich in face value cards, they
are more likely to bust. The player does not have to hit that hand where the dealer does.
Also, a hand of 9, 10, or 11 allows you to double down on a ten-rich deck.
v. B.
-
Hi-Lo Count
There are many methods to count cards. One basic method involves a strategy
where one calculates an advantage/disadvantage based on the number of lO-value cards
and Aces vs. the cards with numeric values between two and six. If the number of tenvalued cards and Aces in the deck is low, then the player will modify his doubling down
and splitting decisions. Card counters use a betting system of betting their smallest
wagers when the deck is rich in low valued cards, since his chances of winning in this
case are reduced. When the deck is rich in high cards the card counter will increase his
original bet since his chances of winning are now increased.
Tracking cards takes experience and patience. One of the most popular counting
systems in use today is the hi-Io system. This system assigns a value of + 1,0, or -1 to
every card dealt at the table, including those dealt to the dealer. Aces and ten valued
-
cards are assigned a value of minus 1. A value of zero is assigned to sevens, eights, and
20
nines and cards two through six each count as a + 1. As the cards are dealt, the counter
keeps a mental tally on the total value of all cards dealt. He makes wagering decisions
based on the current count total. This total is usually adjusted by dividing it by the
number of decks left. A count from a deck that is down to 25 cards is certainly more
credible than a deck that has 250 remaining and this adjustment takes care of this
problem. This adjusted count is known as the true count. One recommendation for the
size of the bets with respect to the true count is shown below:
True Count
Less than 1
1 to 3
3 to 5
5 to 7
Greater than 7
Bet
Minimum Bet
Minimum Bet
Minimum Bet
Minimum Bet
Minimum Bet
*2
*3
*4
*5
If the true count total is high, then the deck is rich in tens and Aces. This is when
the counter will make his biggest bets. If the running true count is around zero, then
neither the player nor the dealer has the advantage. The higher the minus count, the
greater the advantage is to the dealer. This is because the player will be dealt more stiff
hands (hand totals of 12-16). It also means the dealer will bust more so the player may
choose to use this to his advantage (perhaps in a doubling down situation on hands that he
would not normally double down on).
As the dealing progresses, the credibility of the count becomes more accurate.
The size of a players wager then could be increased or decreased with a higher
probability of winning or losing based on the count. A player's decision process is still
based on Basic Strategy. Occasionally, a player may choose to stray from Basic Strategy
-
based on the count. One example of this is a hard 16 versus the dealers 10. If the
21
-
running count is high in this situation, it is better suited to stand instead of hit, which is
what Basic Strategy tells the player to do. This is because the deck is rich in tens and the
player is more likely to bust.
V. C. Core Strategy
Scott Frank introduced another popular counting strategy. He developed a
strategy that gives the player a better chance of being dealt a 19, 20, 21, or blackjack. It
also gives the player a higher chance of getting a nine or 10 if he doubles down. With his
strategy is a higher chance of busting as your means of losing, but not a higher chance of
losing. Frank believed that keeping track of fives, sixes, sevens, and eights, or the core
cards, could be very advantageous to the player. The core strategy was designed because
-.
he thought that stiff hands decided the game. Whether or not to hit on 12-16 is the
toughest decision in blackjack. It is the decision that will make or break a player's game.
Knowing whether or not the deck is heavy in the core cards will help a player make their
decision.
There are 169 possible combinations a player can have with his first two cards.
An active hand is one in which the total value is less than 17, and there are 114 active
hands (including soft hands). Of the 114, 70 of them, or 61 %, include core cards. There
are 21 hard double hands (nine,lO, or 11), 19 of these hands include core cards. About
38% of the hands a player is dealt will be stiffs. The core strategy is supposed to make
these stiffs easier to deal with.
Keeping track of these core cards is easier than tracking lOs using the hi-Io
method. This is because the counters are only keeping track of four cards instead of 10.
22
-
Frank recommends counting core cards by looking at how many have been played
compared to how many quarter decks have been played. This is best explained by
considering an example. One would expect to see 32 core-cards if two decks had been
played (eight quarter decks). The expectation comes from the assumption that each
quarter of a deck will yield four core cards. If a player only sees 24 core-cards, then his
count would be -8. This leaves him with a heavy core shoe. Basically, the player is
comparing what his count should be with what it actually is. On the contrary, a high
positive count means it is a light core shoe.
Heavy core and light core strategies are completely different. This is because
they are two completely different styles of games. A heavy core shoe will be a slower
game. There will be a lot of two card hands with a value of 10-18. There will be double
-
down opportunities on nine, 10, and 11, but they will probably take fewer successful hits.
Hitting an 11 in a heavy core deck would likely end up a 16 or 17, less than the desired
18.5. Overall there will be less busting in a heavy core game. 12-14s though will take
more successful hits. There will also be fewer blackjacks, which is a disadvantage to the
player. Most player losses will be by stiffs left standing or with standing hands one or
two points lower than the dealer.
A light core shoe is better statistically for the player. The player will be dealt
more 19s, 20s and blackjacks. There will not be as many double down opportunities, but
those opportunities will statistically take better hits. Busting activity will increase
overall. The dealer will bust on more stiffs, while the player will take less successful 1214 hits. The increased number of blackjacks in this type of deck will help the player.
-
Most player losses will be by busting or by standing hands two or more points lower than
23
-
the dealer. This is because the player will stand on more 12-14 valued hands if he
follows Scott's strategy. The charts for the core strategy are towards the end of the paper.
The core-card counter seems to have an advantage over the normal 10trackerlbasic-strategy player. The objective of the normal player is to get high-valued
hands. Card counters count in order to have a better chance of being dealt a 20 or 21.
But the core player believes that high-valued hands will play themselves. Their main
objective is to play stiffs and lower hands well. Stiffs and lower hands make up most of
the game. Many lOs in the deck will cause the dealers busting rate to go up, but when the
dealers busting rate goes up, so does the players. If there are a lot of lOs in the deck, then
there will be a lot of 20s, which will cause pushes with the dealer. On the outside, this
would not seem to accomplish very much. The core card count is much easier to track
since there are fewer cards to keep track of.
VI. Set up of the program
The program was set up to mimic a typical game of blackjack that one might find
at a North American casino. The program was done using a Macro in Visual Basic
through Microsoft Excel. The assumptions that are utilized for the program are as
follows:
1) A cut card is randomly inserted into the deck between the 60 th percentile of
th
the deck and the 90 percentile of the deck.
2) The dealer stands on soft 17.
3) All players at the table play the same strategy.
,-
4) In the Martingale strategy, a player quits when one of the following happens:
a. A player does not have enough money for the required bet according
to the strategy.
24
.-
b.
c.
d.
e.
The
The
The
The
player's bankroll hits at or above designated ceiling .
player's bankroll hits at or below a designated floor.
player has outplayed his maximum number of trials.
player's required bet is above the maximum at the table.
5) A player in the Martingale strategy will only quit on a win if none of the
above is satisfied.
6) In multi-player games, Martingale players will leave the table if they have hit
one of the above stipulations.
7) Card counting players use the same strategy as Basic Strategy along with the
hi-Io count.
8) Basic strategy players and card counting players only quit when they do not
have enough money or they have played the designated number of hands.
9) The deck is shuffled by generating random numbers for three columns in
Excel for every card that is located in a freshly used deck. The cards are then
assorted, in ascending order, by the columns in their respective order. This
new deck is the one that is used all the way up to, but not including, the cut
card.
10) A fresh deck is shuffled before each new trial.
11) Players at the table always make the correct decision according to their
strategy.
12) Players always playa set number of hands, with the exception of busting, in
Basic Strategy and card counting.
13) For each trial a player begins with his initial bankroll.
14) A player can double down on any two cards.
15) A player can split any two cards.
16) There is no re-splitting of cards.
17) After a split, an Ace-Ten hand is valued as 21 and only pays out 1: 1.
18) The strategy can change according to the number of decks being played.
19) A player can double down after a split.
25
VII. Analysis
VII. A. Multiple Deck Games
Most of the up-to-date statistics on the player's advantage have come from the
Internet. One of the most useful websites found was Blackjacklnfo.com by Ken Smith.
The statistics given on the website were produced using all of the same assumptions that
were used in this simulation with one major exception. The lone difference in
simulations was that Smith did not assume there was a cut card in the deck. The strategy
charts for Basic Strategy were also the same as those found on his website. These charts
were produced through combinatorics. This website has been around since 1996 and the
numbers posted have been verified by many others in the world of blackjack.
-
Below is the chart produced using the website's Basic Strategy Engine.
Basic Strategy Disadvantages for One Player
Number of Decks
1
2
3
4
5
6
7
Disadvantage
-0.15%
0.20%
0.30%
0.38%
0.41%
0.44%
0.46%
0.47%
8
The chart shows that as one increases the number of decks, the player's disadvantage
grows. Another thing to observe is that with the correct Basic Strategy a player can have
the advantage in a one-deck game. This is why it is extremely rare to find a one-deck
game with the same rules as those commonly found in Las Vegas casinos and the rules
used in this particular simulation.
26
-
Our simulation yielded similar results for multiple deck games. Even though we
put in a cut card, we expected to get approximately the same results Smith calculated in a
Basic Strategy game. Running a simulation of 1000 hands 1000 times for Basic Strategy
gave us approximately the numbers we were looking for. On the following page is a
chart of our findings compared to Smith's findings.
Basic Strategy Disadvantages for One Player
-
Number of Decks
Smith
Our Program
Lower Bound
Upper Bound
1
2
3
4
5
6
7
8
-0.15%
0.20%
0.30%
0.38%
0.41%
0.44%
0.46%
0.47%
-0.24%
0.26%
0.16%
0.26%
0.36%
0.48%
0.48%
0.48%
-0.43%
0.06%
-0.04%
0.06%
0.17%
0.29%
0.28%
0.29%
-0.04%
0.46%
0.36%
0.45%
0.56%
0.68%
0.68%
0.68%
The Smith column of the above chart has the same house advantages as the
previous chart, and the Our Program column has the house advantages calculated from
our simulation of Basic Strategy. A 95% confidence interval gave us a margin of error
by using the standard deviation calculated from the number of trials. The two columns to
the right on the graph above are the upper and lower bounds of this confidence interval.
These numbers tell us that we are 95% sure the true advantage falls within the bounds.
Since Smith's results fall within our confidence interval, we cannot assume the cut card
made a difference in the results.
Had we simulated more than 1000 trials, we may have been able to shorten the
confidence intervals. Because our confidence intervals are so wide it is not conclusive
that an increase in decks causes a decrease in the player advantage for a player using
27
-
Basic Strategy. Even though our evidence is not conclusive, our expected winning did
decrease as the number of decks increased. This is the trend that we expected based on
our research. It is much more likely to find multiple deck games than single deck games,
and since casinos are trying to make as much money as they can, it is assumed that their
use of multiple deck games over single deck games gives them a larger advantage.
According to Smith, Las Vegas casinos typically have six deck games. A person
who bets $1000 in one gambling session would be expected to lose $4.40. This is truly
not a substantial amount compared to the total amount bet. At a five-dollar table, playing
40 hands per hour, it would take approximately five hours to bet $1000. On average, a
person will pay $4.40 for five hours of blackjack at the casino using the proper Basic
Strategy.
-
A new trend found in casinos today is the use of a continuous shuffler. The use of
this machine essentially ensures that all cards dealt are from a fresh deck for every round
of the game. This takes away all the advantage in card counting, and it is for this reason
that most people believe that the casinos use this machine. However, this is only part of
the reason. In using this machine, the continuous shuffling machine randomly inserts the
discards from the previous round back into the deck. This speeds up play, and since
blackjack is a negative expectation game, it allows more bets to be laid out on the table,
and thus increases the house take. The chart below, taken from TheWizardofOdds.com,
shows that the house advantage is actually decreased through use of this machine.
28
House Edge with and without Continuous Shuffling Machine (CSM)
Number of Decks
Cut Card
CSM
Difference
1
2
4
6
8
0.16%
0.26%
0.41%
0.45%
0.48%
0.05%
0.20%
0.37%
0.43%
0.46%
0.11%
0.06%
0.03%
0.02%
0.01%
Certainly it seems that a casino would never introduce a machine that significantly
reduces its take, and thus it seems that the continuous shuffling machine does not
significantly reduce the house's overall take, since it is still a positive expectation. It
seems that this machine will be around for a long time since it is an easy way to take
away the advantage of a card counter while the house still retains its advantage.
VII. B. Common Human Errors
Players often make bad assumptions about the game. All too often a player gets
frustrated with a bad run and wants to make up for past losses, so they increase their bet
size. The assumption that a player can make up for a string of bad luck with a larger bet
size only increases the expected loss, since more money is laid out in a negative
expectation game. Frequently players base their decision on how lucky they feel. A
typical situation is when a player has a hard 16 versus a lO-valued upcard for the dealer.
According to all Basic Strategy charts a player should hit in this situation, but often a
player will choose to stand. The player's rationale is formed from their run of "bad
luck", the other cards on the table, or the card that the previous player took last.
,-
However, this does not follow the strategy that is founded on the mathematics of the
29
-
game. Veering from the strategy increases the expected loss of the player and thus gives
him less of a chance to win.
The expectation of a player's final bankroll is also lowered because people make
mistakes. A beginner will, on average, make more mistakes than an experienced player,
but mistakes are common in long sessions for any experience level. People get tired,
distracted, or even intoxicated from free drinks given out by casinos. This makes a
player less likely to make the proper play, and this lowers his expected gain.
Another reason that people often lose is that players often become greedy. When
players are winning they want to win more and do not realize that they might be at their
highest winning level over the long run. Commonly, people lose all of their winnings
and then continue to lose in order to get back to that maximum level. This combination
.-
ultimately produces large losses, largely due to players making proportionately larger
bets in order to make up for their losses.
Another problem that a casino gambler often has is that he will often be tight with
his money when he is losing, and loose when he is winning. The only way to keep the
house advantage minimal is for the player to stick to his strategy. A player cannot veer
from this strategy. Using the money loosely will often imply that the player is taking
unneeded risk in his bets by either increasing the bet or by making the wrong decision.
The tight player probably does worse since this player will not take advantage of double
down and split opportunities, which are the best times to have more money on the table.
This implies that a player needs a large bankroll that is funded by money that the player
does not care to lose.
30
-
These four problems produce a much lower expected value for the player. The
extent is unknown since the level of mistakes is different for each respective player, but it
is certainly significant. This improper play is one reason why the casino takes such a
large cut from the amount bet in blackjack. It would seem that casinos would worry
about losing this increased advantage, but they are not too worned about it since players
will still have a negative expected value.
Casinos know their cut would be reduced, but still do not worry about players
learning how to play the proper way. The small advantage that casinos have turns into
big profits with so much money being wagered on blackjack. According to
Casinocenter.com, blackjack is the most popular casino game. Casinos often offer free
blackjack training sessions and some even allow the use of a strategy card at the table.
-.
However, a lot of money is still spent on the psychology of gambling. Everything in a
casino is planned from the placement of blackjack tables to the color of the carpet on the
casino floor. This suggests that the casino still wants the advantage they receive from
human error and natural tendencies, so it is debatable as to whether casinos really want to
help the player. It would seem that a casino would like to lower a player's losses so that
they would want to come back again, but they do not want to eliminate the losses so
much that the house loses.
VII. C. Results
With the assumptions in mind the program produced some expected and some
unexpected results. Basic strategy yielded a greater disadvantage for the player than card
counting did. Card counting yielded higher results for all of the simulations. The chart
shows that card counting gives a better expectation for a player in a one-deck game and a
31
six deck game. Since the confidence intervals do not overlap for both deck games, it is
assumed that card counting produces higher results.
Comparison of Player's Disadvantage for Basic Strategy vs Card Counting
Number of Decks
1
6
BS
-0.24%
0.48%
BSlB
-0.43%
0.29%
BSUB
-0.04%
0.68%
CC
-0.56%
0.08%
CClB
CCUB
-0.65%
0.02%
-0.47%
0.15%
Another expectation that seemed to be true was that having mUltiple players at the
table did not change the odds significantly. The chart below does not show any
significant difference between any of the players. All the results appear to have the same
trend and it is therefore assumed that they are all as likely.
Frequencies for 6P 60 BS
140.-----------------------------------------.
120 + - - - - - - - - - ;
100 +-------.......,....,
>.
---+- Player 1
80+-----------~
_Player 2
u
c:
~
C'
2!
u..
60+-----------~~~--~'~r------------------~
40+---------~. .r_----~~----------------~
Player 3
~Player4
"""*- Player 5
--.- Player 6
20+-----
O. . . . . . . . .
~~~~~~~
~ ~ ~ ~ ~ ~
~ ~
~
~ ~ ~
~
~ ~ ~
~
~~~~~~~~o/V~~~~~o/~
Ending Bankroll
The expectation that card counting would be much more effective for one-deck
games as opposed to multiple deck games also seemed to be true. As is shown in the
32
chart below, the frequencies of higher bankrolls are higher for one-deck games than for
six-deck games since the one-deck chart is shifted to the right of the six-deck chart.
Comparison of 1D vs. 6D CC
250
200
>(.)
c 150
CI)
:::I 100
0"
...CI) 50
LL
0
-50
~1
Deck
- 6 Deck
a
T""
T""
T""
M
~
T""
T""
~
00
T""
T""
0
N
N
N
~
N
~
a
~
~
~
m
N
N
N
Ending Bankroll
-
We also expected that the variance would be greatest for the Martingale, then for
the card count, then for the
B~sic
Strategy. The Martingale chart does not seem to
compare easily to the card counting and the Basic Strategy charts since it has such a high
variance. The six-deck card counting is compared to a six-deck Basic Strategy run in the
first chart below while the second chart has a Martingale betting strategy with a ceiling of
$5,500 and a floor of$3,000.
33
,---------------------~---"-"-
Comparison of 60 CC vs 60 BS
800 , - - - - - - - - - - - - - ,
700
>. 600
g 500
I • Basic Strategy
~ 400
Card Counting
g
...u. 300
200
100
'i
0_.
-
~C) ~C) ~C) <:>\) ~C) ~C) ~C) <:>C) ~C)
"f); ,,~ ~
f).,f);
~ ~<:s
r65
,,<:s
{3
Ending Bankroll
Ending Bankroll for Martingale
5500 Ceiling, 3000 Floor
700
600
500
>.
(J
c 400
CD
::I
300
C'"
...CD 200
LL
100
0
-100 ~~
0
N
i
L_
:-
....
LO
co
N
2
('f)
('f)
LO
\.,.
0
LO
2
"~
r--::
-.::t -.::t LO co co
Ending Bankroll
0
I'--
('f)
34
0
~
~
I'--
LO
0
IX)
Frequency
-
The charts show the variance for Basic Strategy is the lowest, followed by card counting,
and then Martingale. This comes as no surprise since Basic Strategy has the most steady
bet sizes. As expected, the results for the Hi-Lo card count betting strategy and Basic
Strategy appeared to be approximately Normal, while the results for the Martingale
betting strategy were skewed to the right. Since everyone quits after a certain level in the
Martingale, all results in the trials would be wins or busts. The only variability in the
simulation comes from the size of the final bet. Because Basic strategy and card counting
run a specific number of trials each time, the ending bankroll becomes Normal.
The Martingale strategy produced an expected result. It seems that only playing
the Martingale for small amounts gives a higher expectation than for medium amounts as
a ceiling. This shows that being humble in one's winnings will produce winnings more
often. The fact that the average ending bankroll is nearly the original value of $5,000
shows that there is a significant risk with this strategy and one should use caution in using
this. Since the Martingale betting strategy has such a high variance, however, this
conclusion is hard to justify with the capabilities of the program. Certainly having one or
two more bust outs instead of winners would have had a significant effect on the average
bankroll.
Martingale Results with Different Ceiling Levels
Floor
Ceiling
3000
3000
3000
3000
5250
5500
5750
6000
Expected
Endi~ Bankroll
5010.0675
4977.36125
4994.4925
4986.69
The frequencies of the number of trials that hit the ceiling did not come as a
--
surprise. The higher the ceiling, the less likely it seems that it is to hit that value before
35
hitting the floor. This further shows that using a statistically unfounded betting strategy
and becoming overzealous in one's efforts will eventually lead to ruin in the long run.
4 Different Ceiling Maxes
1000
800
~
u
c:
CI)
:::l
C"
...
LL
-5250
-5500
5750
-6000
600
400
CI)
200
~
0
-200
~~
I
()
50
"'-
100
1~) o
Ending Bankroll
VII. D. Online Blackjack
The rapid growth in online gambling has produced what seems to be a wonderful
opportunity for blackjack players. Many online casinos offer "free" money to new
players registering at their casinos. This "free" money is subject to the terms and
conditions of the specific casino, but is typically offered in the following manner: a
player gets a 100% match bonus for every dollar deposited at the casino up to a set
amount (typically between $50 and $150). The player earns the money in the following
fashion. A player must play the amount of the bonus plus the initial deposit a certain
.-.
number of times (typically four to eight) to earn the deposit bonus. Once this is done a
36
player can withdraw their bankroll and has a full claim to the bonus. Withdrawals prior
to hitting this level are prorated for the bonus amount (or sometimes no bonus is given).
This situation lends itself to a great opportunity for an intelligent blackjack player.
Typically the rules in the casino are less favorable to the player than normal (such as
having a dealer hit on soft totals of 17 when necessary), but the house advantage is
usually nominal. If a player deposits $150, he will then receive $150 as a bonus. If the
requirement is for a player to play six times this total, he will have to play $1800 worth of
hands. Assuming the house has an edge of around .5% the player is expected to lose $9
($1800
* .005).
However the player will have received $150 as a deposit bonus, netting a
gain of $141. It appears that it would be extremely unlikely that a player would even lose
in this situation with such a small disadvantage, and this seems to be a great opportunity
to take a small risk to get a sizeable gain.
This offer would seem to be a bad business decision for the casino, but there are
hundreds of casinos that offer a similar sign-up program. Internet casinos seem to be
taking all the money that they save by having a casino on the Internet, and using it to
finance special offers. Certainly an online casino has costs, but they are nothing like
what it costs to run a real casino. According to Casinogenerator.com, an online casino
could be set up for a total as low as $5000, far less than the millions a normal casino
would cost to set up. This gives the house more money to payout free amenities to
entice new players to get involved in online gambling with their casino. An online casino
might expect to have this offer generate profits in the future from a new player, or that the
-.
new player will play overly risky with the free money, or that a player will not live up to
the terms and conditions of the deposit offer. This deposit bonus seems to be a great
37
opportunity for the player, but the risks outside of the gambling risk must also be
assessed before one jumps into online gaming.
The outside risks involved make this offer much more risky than it might seem.
A player must take into account the reputation of the online casino, how easy it will be to
get the money, the risk of passing confidential information on the Internet, and the
increased ability to become addicted to gambling. Knowing that a casino has paid out
well to other players should be an important part in evaluating a deposit bonus offer. If it
is known that an online casino has been troublesome for players to deal with in payouts,
it should not be used. A player still needs to make sure he is willing to accept all of these
risks and that this offer is from a reputable casino. The informed player who researches
the casino and can play with money that he can afford to lose, should be able to take
advantage of this opportunity, which will probably disappear in a few years. A player
should find out how long it typically takes to get the money, what restrictions there are on
withdrawals, and should also be able to contact the casino by phone. With research, a
player can take advantage of this lucrative opportunity, but must enter into it with
caution.
VIII. Conclusion
The optimal strategy in blackjack cannot be the one that goes against the
mathematics behind the game. Many betting and card strategy systems may seem to be
better than those developed before, but they will not be able to produce better results than
card counting in the long run. The only way to preserve the optimal strategy is to stick
with this strategy throughout the game. To do this, a player must have the patience, the
38
bankroll, and the knowledge that a loss is likely in this game. Understanding the game
and the mathematics behind the game is the first step to minimizing the house's
advantage and eliminating the costly mistakes that a player often makes in a casino.
Combining this knowledge with some situations where the player has an advantage, such
as the new bonus in online blackjack, bring the possibility of using this knowledge for
profit.
39
IX. Strategy Charts
IX. A. Basic Strategy Charts
The first chart for each corresponding number of decks is the chart by which one
decides whether or not to split a pair in regards to the dealer's upcard. The next chart in
each respective section is the chart by which one makes a decision based on a player's
hard total to hit, double down, or stand against the dealer's upcard. The last chart in each
respective section is the chart by which one makes a decision based on a player's soft
total to hit, double down, or stand against the dealer's upcard. The charts below are keys
for each of the three respective charts. All Basic Strategy charts are taken from
Blackjacklnfo.com.
Split Chart Key
y
N
S
Split
Do Not Split
Stand
Hard Total Key
H
S
0
Hit
Stand
Double Down if possible, otherwise hit
OS
Double Down if possible, otherwise stand
S
0
Soft Total Key
Hit
Stand
Double Down if possible, otherwise hit
OS
Double Down if possible, otherwise stand
H
40
IX. A. 1. One Deck Charts
Split Chart for One Deck
Dealer's Upcard
Pairs
Cards
Ace, Ace
2
Y
3
Y
4
Y
5
Y
6
Y
7
Y
8
Y
9
Y
T
Y
Y
10, 10
9,9
8,8
7, 7
6,6
5,5
4,4
3,3
2,2
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
Y
Y
Y
Y
Y
N
N
N
N
Y
S
Y
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
Y
Y
N
N
N
Y
N
N
N
N
N
N
N
A
N
N
N
N
N
N
N
N
N
N
N
Hard Total Chart for One Deck
Hard
Totals
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Dealer's Upcard
7
S
S
S
S
S
8
S
S
S
S
S
9
S
S
S
S
S
T
A
S
S
S
S
S
S
S
S
S
S
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
D
H
H
H
H
H
H
H
H
H
H
D
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
4
S
S
S
S
S
S
S
S
S
S
5
S
S
S
S
S
S
S
S
S
S
6
S
S
S
S
S
S
S
S
S
S
D
D
D
H
H
H
D
D
D
D
H
H
D
D
D
D
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
2
S
S
S
S
S
S
S
S
S
3
S
S
S
S
S
S
S
S
S
H
D
D
D
H
H
H
H
D
D
D
H
H
H
H
H
41
Soft Total Chart for One Deck
Soft
Totals
Dealer's Upcard
Cards
2
3
4
5
6
7
Ace + 10
Ace+9
Ace+8
Ace+7
Ace +6
Ace+S
Ace+4
Ace+3
Ace+2
Ace + 1
S
S
S
S
S
S
S
S
S
S
S
DS
S
DS
S
S
DS
DS
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
S
H
H
H
H
H
H
S
S
S
D
DS
D
H
H
H
H
H
H
H
H
H
H
D
42
S
8
S
S
S
S
H
H
H
H
H
H
9
T
A
S
S
S
S
S
H
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
S
S
S
H
H
H
H
H
H
IX. A. 2. Two Deck Charts
Split Chart for Two Decks
Dealer's Upcard
Splits
Cards
Ace, Ace
10,10
9,9
8,8
7, 7
6,6
5,5
4,4
3,3
2,2
2
Y
N
Y
Y
Y
Y
N
N
Y
Y
3
Y
N
Y
Y
Y
Y
N
N
Y
Y
4
Y
N
Y
Y
Y
Y
N
N
Y
Y
5
Y
N
Y
Y
Y
Y
N
Y
Y
Y
6
Y
N
Y
Y
Y
Y
N
Y
Y
Y
7
Y
N
N
Y
Y
Y
N
N
Y
Y
8
Y
N
Y
Y
Y
N
N
N
N
N
9
Y
N
Y
Y
N
N
N
N
N
N
T
Y
N
N
Y
N
N
N
N
N
N
T
S
S
S
S
S
H
H
H
H
H
A
Y
N
N
Y
N
N
N
N
N
N
Hard Total Chart for Two Decks
Hard
Totals
-
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Dealer's Upcard
2
3
4
5
6
7
8
9
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
H
D
D
D
H
H
H
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
H
D
D
D
H
H
H
S
S
S
S
S
S
S
S
S
S
D
D
D
D
D
H
H
H
H
H
H
H
H
D
D
H
H
H
H
D
D
H
H
H
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
H
H
H
H
D
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
D
D
43
D
H
A
S
S
S
S
S
Soft Total Chart for Two Decks
Soft
Totals
Cards
Ace + 10
Ace +9
Ace+8
Ace+7
Ace+6
Ace+5
Ace+4
Ace+3
Ace + 2
Ace + 1
Dealer's Up_card
2
3
4
S
S
S
S
S
S
S
S
S
S
DS
DS
H
H
H
H
H
H
D
H
H
H
H
H
D
D
D
H
H
H
5
S
S
S
DS
D
D
D
D
D
D
6
7
S
S
S
S
S
S
DS
S
D
D
D
D
D
D
44
H
H
H
H
H
H ,
8
S
S
S
S
H
H
H
H
H
H
9
T
A
S
S
S
S
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
IX. A. 3. Three Deck Charts
Split Chart for Three Decks
Dealer's Upcard
Pairs
Cards
Ace, Ace
10,10
9,9
8,8
7, 7
6,6
5,5
4,4
3,3
2,2
.-
4
Y
5
Y
6
Y
N
N
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
7
Y
N
N
Y
Y
N
N
N
Y
Y
8
Y
9
Y
N
N
Y
Y
Y
Y
N
N
N
N
N
N
N
N
T
Y
N
N
Y
N
N
N
N
N
N
N
N
N
N
A
Y
N
N
Y
N
N
N
N
N
N
Hard Total Chart for Three Decks
Hard
Totals
Dealer's Upcard
Total
2
3
4
5
6
7
8
9
T
21
20
19
18
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
0
0
0
0
0
0
0
0
0
0
0
H
H
H
H
D
H
H
H
S
S
S
S
S
S
S
S
S
S
0
0
0
S
S
S
S
S
H
S
S
S
S
S
S
S
S
S
S
0
0
S
S
S
S
S
H
S
S
S
S
S
S
S
S
S
S
0
0
0
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
17
16
15
14
13
12
11
10
9
8
7
--
3
Y
2
Y
6
5
4
3
2
H
H
H
D
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
A
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
45
Soft Total Chart for Three Decks
Soft
Totals
Dealer's Upcard
Cards
2
3
4
5
6
7
8
9
T
A
Ace + 10
S
S
S
S
S
S
S
S
S
S
Ace + 9
S
S
S
S
S
S
S
S
S
S
S
DS
S
DS
S
DS
S
DS
S
S
S
H
S
H
H
H
D
H
H
D
D
D
D
D
H
H
D
D
D
H
H
H
H
H
H
H
H
H
H
H
D
D
H
S
S
H
H
S
S
H
S
Ace + 8
H
H
H
H
Ace +7
Ace +6
Ace +5
Ace+4
Ace+3
Ace+2
Ace + 1
H
H
H
H
H
H
H
H
H
D
D
D
D
D
46
H
H
H
H
H
H
H
H
H
H
H
IX. A. 4. Four Deck Charts
Split Chart for Four Decks
Dealer's Upcard
Pairs
Cards
2
3
4
5
6
7
8
9
Ace, Ace
Y
N
Y
Y
N
Y
Y
N
Y
Y
N
Y
Y
N
Y
Y
N
N
Y
N
Y
Y
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
N
Y
Y
N
N
N
N
N
N
N
N
N
N
N
N
10,10
9,9
8,8
7, 7
Y
Y
Y
Y
Y
6,6
5,5
4,4
3,3
2,2
Y
Y
Y
Y
Y
N
N
Y
Y
N
N
Y
Y
N
N
Y
Y
N
Y
Y
Y
N
Y
Y
Y
T
Y
N
N
Y
N
N
N
N
N
N
A
Y
N
N
Y
N
N
N
N
N
N
Hard Total Chart for Four Decks
Hard
Totals
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Dealer's Upcard
2
S
S
S
S
S
S
S
S
S
3
S
S
S
S
S
S
S
S
S
H
0
0
H
H
H
H
H
0
0
0
H
H
H
H
H
H
H
H
H
H
H
4
S
S
S
S
S
S
S
S
S
S
5
S
S
S
S
S
S
S
S
S
S
6
S
S
S
S
S
S
S
S
S
S
0
0
0
H
H
H
H
H
0
0
0
H
H
H
H
H
9
S
S
S
S
S
T
S
S
S
S
S
8
S
S
S
S
S
0
0
0
H
H
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
H
H
0
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
A
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
47
7
S
S
S
S
S
Soft Total Chart for Four Decks
Soft
Totals
Cards
Ace + 10
Ace +9
Ace +8
Ace+7
Ace+6
Ace+S
Ace+4
Ace+3
Ace+2
Ace + 1
Dealer's Upcard
2
3
4
5
6
7
8
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
OS
OS
OS
OS
S
S
H
H
H
H
H
0
H
H
H
0
0
0
0
0
0
H
H
0
0
0
0
0
0
H
H
H
H
0
0
0
H
H
H
H
48
H
H
H
H
H
H
H
H
9
S
S
S
H
H
H
H
H
H
H
T
A
S
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
IX. A. 5. Five Deck Charts
Split Chart for Five Decks
Pairs
Cards
Ace, Ace
10, 10
9,9
8,8
7, 7
6,6
5,5
4,4
3,3
2,2
2
Y
N
Y
Y
Y
Y
N
N
Y
Y
3
4
5
Y
N
Y
Y
Y
Y
N
N
Y
Y
Y
N
Y
Y
Y
N
Y
Y
Y
Y
N
Y
Y
Y
Y
Y
N
N
Y
Y
Dealer's Upcard
7
6
Y
N
Y
Y
Y
Y
N
Y
Y
Y
Y
N
N
Y
Y
N
N
N
Y
Y
8
9
T
A
Y
N
Y
Y
N
N
N
N
N
N
Y
N
Y
Y
N
N
N
N
N
N
Y
N
N
Y
N
Y
N
N
Y
N
N
N
N
N
N
N
N
N
N
N
Hard Total Chart for Five Decks
Hard
Totals
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Dealer's Upcard
7
6
4
S
S
S
S
S
S
S
S
S
S
5
S
S
S
S
S
S
S
S
S
S
D
D
D
D
D
D
D
H
H
H
D
H
H
H
H
H
H
H
H
H
H
H
2
S
S
S
S
S
S
S
S
S
3
S
S
S
S
S
S
S
S
S
H
D
D
H
H
H
H
H
D
D
D
H
H
H
H
H
H
H
8
S
S
S
S
S
9
S
S
S
S
S
D
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
D
D
H
H
H
H
H
H
H
H
H
D
D
H
H
H
H
H
H
D
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
A
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
S
S
S
S
S
S
S
S
S
S
49
S
S
S
S
S
H
H
H
T
S
S
S
S
S
H
H
H
Soft Total Chart for Five Decks
Soft
Totals
Dealer's Upcard
Cards
2
3
4
5
6
7
8
9
T
A
Ace + 10
S
S
S
S
S
S
S
S
S
S
S
S
S
S
Ace+9
Ace+8
Ace+7
S
S
S
S
S
S
DS
S
DS
S
DS
S
DS
S
H
S
H
S
H
D
D
D
D
S
S
H
H
H
H
H
H
D
D
H
D
D
D
D
H
H
H
D
H
H
D
D
D
H
H
H
H
H
H
H
D
H
H
H
H
H
H
H
Ace+6
Ace+5
Ace+4
Ace + 3
Ace +2
Ace + 1
S
H
H
H
H
H
H
H
H
H
S
D
50
S
S
S
H
H
H
H
H
H
H
H
H
IX. A. 6. Six Deck Charts
Split Chart for Six Decks
Dealer's Upcard
Pairs
Cards
Ace, Ace
10,10
9,9
8,8
7, 7
6,6
5,5
4,4
3,3
2,2
2
Y
N
Y
Y
Y
Y
N
N
Y
Y
3
Y
N
Y
Y
Y
Y
N
N
Y
Y
4
Y
N
Y
Y
Y
Y
N
N
Y
Y
5
Y
N
Y
Y
Y
Y
N
Y
Y
Y
6
Y
N
Y
Y
Y
Y
N
Y
Y
Y
7
Y
N
N
Y
Y
N
N
N
Y
Y
8
Y
N
Y
Y
N
N
N
N
N
N
A
9
Y
N
Y
Y
N
N
N
N
N
N
T
Y
N
N
Y
N
N
N
N
N
N
Y
N
N
Y
N
N
N
N
N
N
Hard Total Chart for Six Decks
Hard
Totals
5
6
7
8
9
T
A
S
S
S
S
S
S
S
S
S
S
D
D
0
S
S
S
S
S
S
S
S
S
S
D
D
0
S
S
S
S
S
S
S
S
S
S
D
D
0
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
3
S
S
S
S
S
S
S
S
S
H
D
D
0
H
H
H
H
H
H
5
4
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
3
2
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
.-
Dealer's Upcard
4
2
S
S
S
S
S
S
S
S
S
H
D
D
H
H
H
H
H
H
51
H
H
H
H
H
H
H
H
H
H
H
H
D
D
D
0
D
0
D
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
Soft Total Chart for Six Decks
Soft
Totals
Cards
Ace + 10
Ace +9
Ace+8
Ace+7
Ace+6
Ace+5
Ace+4
Ace+3
Ace+2
Ace + 1
Dealer's Upcard
2
3
4
5
S
S
S
S
S
S
S
S
S
S
S
S
S
H
H
H
H
H
H
OS
OS
OS
6
S
S
S
OS
D
D
D
D
0
D
0
D
D
D
D
D
D
D
D
D
H
H
H
H
H
H
H
H
52
7
S
S
S
S
H
H
H
H
H
H
8
S
S
S
S
H
H
H
H
H
H
T
A
S
S
S
S
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
9
IX. A. 7. Seven Deck Charts
Split Chart for Seven Decks
Dealer's Upcard
Pairs
Cards
2
3
4
5
6
Ace, Ace
Y
N
Y
N
Y
Y
N
Y
Y
N
Y
Y
N
Y
Y
Y
Y
6,6
5,5
4,4
3,3
2,2
Y
Y
Y
Y
Y
Y
7, 7
Y
Y
Y
Y
Y
10, 10
9,9
8,8
N
N
N
N
Y
Y
Y
Y
N
N
Y
Y
Y
Y
N
N
Y
Y
Y
Y
Y
Y
7
Y
N
N
Y
Y
N
N
N
Y
Y
8
9
Y
N
Y
Y
Y
N
N
N
N
N
N
N
Y
Y
N
N
N
N
T
Y
N
N
Y
N
N
N
A
Y
N
N
N
N
N
N
N
Y
N
N
N
N
N
N
Hard Total Chart for Seven Decks
Hard
Totals
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Dealer's Upcard
7
S
S
S
S
S
8
S
S
S
S
S
9
S
S
S
S
S
T
A
S
S
S
S
S
S
S
S
S
S
H
H
H
0
0
0
H
H
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
H
H
0
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
4
S
S
S
S
S
S
S
S
S
S
5
S
S
S
S
S
S
S
S
S
S
6
S
S
S
S
S
S
S
S
S
S
0
0
H
H
H
0
0
0
H
H
H
H
H
H
H
H
H
H
H
2
S
S
S
S
S
S
S
S
S
3
S
S
S
S
S
S
S
S
S
H
0
0
H
H
H
H
H
0
0
0
0
53
Soft Total Chart for Seven Decks
Soft
Totals
Cards
Ace + 10
Ace+9
Ace+8
Ace+7
Ace+6
Ace+5
Ace+4
Ace+3
Ace+2
Ace + 1
Dealer's Upcard
2
S
S
S
S
H
H
H
H
H
H
3
S
S
S
OS
0
H
H
H
H
H
4
6
7
S
S
S
5
S
S
S
S
S
S
S
S
S
S
OS
S
OS
S
S
OS
S
S
S
H
S
H
S
H
0
0
0
H
0
0
0
0
0
0
0
0
0
0
0
0
0
0
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
54
8
S
S
H
H
H
H
H
H
9
T
A
S
S
S
-
IX. A. 8. Eight Deck Charts
Split Chart for Eight Decks
Dealer's Upcard
Pairs
Cards
2
3
Ace, Ace
Y
N
Y
N
Y
Y
Y
10,10
9,9
8,8
4
5
6
Y
N
Y
Y
N
Y
Y
N
Y
Y
Y
7, 7
Y
Y
Y
6,6
5,5
4,4
3,3
2,2
Y
Y
Y
Y
N
N
Y
Y
N
N
Y
Y
N
N
Y
Y
Y
Y
Y
Y
Y
N
Y
Y
Y
N
Y
Y
Y
7
Y
N
N
Y
Y
N
N
N
Y
Y
8
9
T
A
Y
N
Y
N
Y
Y
N
Y
N
N
Y
N
N
N
N
N
N
N
Y
Y
Y
N
N
N
N
N
N
Y
N
N
N
N
N
N
N
N
N
N
N
N
Hard Total Chart for Eight Decks
Hard
Totals
Total
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
-
6
5
4
3
2
Dealer's Upcard
H
4
S
S
S
S
S
S
S
S
S
S
5
S
S
S
S
S
S
S
S
S
S
6
S
S
S
S
S
S
S
S
S
S
H
H
H
H
0
0
0
0
0
0
0
0
0
0
0
0
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
2
S
S
S
S
S
S
S
S
S
3
S
S
S
S
S
S
S
S
S
H
0
0
9
S
S
S
S
S
T
S
S
S
S
S
8
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
0
0
0
0
0
0
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
A
S
S
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
55
7
S
S
S
S
S
H
H
0
Soft Total Chart for Eight Decks
Soft
Totals
Cards
Ace + 10
Ace+9
Ace+8
Ace+7
Ace+6
Ace+5
Ace+4
Ace+3
Ace+2
Ace + 1
Dealer's Upcard
2
3
4
5
6
7
8
9
T
A
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
DS
S
DS
S
DS
S
DS
S
S
S
S
H
S
H
S
H
D
H
D
D
D
D
D
D
D
H
H
H
H
H
H
H
H
H
H
H
D
D
D
D
H
H
D
D
D
D
H
H
H
H
H
H
H
H
H
H
H
H
S
H
H
H
H
H
H
H
H
H
H
56
S
H
H
H
H
H
H
H
H
IX. B. Core Strategy Charts
Core Strategy charts taken from Blackjack For Winners
Core Strategy Key
Hit
Stand
Double Down
Split
Do Not S~it
H
S
D
y
N
Note: For the Hard Totals, Numbers indicate at what count to change from the normal
strategy, H becomes S, S becomes H.
Hard Totals
Normal Strategy
Dealer's Upcard
Player total
17
16
15
14
13
12
2
S
S
S
S
S
H
3
S
S
S
S
S
H
4
S
S
S
S
S
S
5
S
S
S
S
S
S
7
S
H
H
H
H
H
6
S
S
S
S
S
S
8
S
H
H
H
H
H
9
S
H
H
H
H
H
10
S
H
H
H
H
H
A
10
S
-1
-4
-5
-7
A
S
-14
-10
-9
-10
S
H
H
H
H
H
Light Core Strategy
Dealer's Upcard
Player total
17
16
15
14
13
12
2
S
S
S
S
S
-3
3
S
S
S
S
S
-2
4
S
S
S
S
S
5
S
S
S
S
S
6
S
S
S
S
S
7
S
H
H
H
-14
S
H
H
H
-14
9
S
-16
-12
-10
-10
S
S
S
H
H
H
-14
-15
8
9
S
16
H
H
H
H
10
S
H
H
H
H
H
A
Heavy Core
Strate~
Dealer's
-
Player total
17
16
15
14
13
2
S
S
12
H
8
4
1
3
S
S
11
6
2
H
4
S
S
S
8
4
1
5
S
S
S
12
6
2
57
8
6
S
S
S
14
6
2
U~card
7
S
8
H
H
H
H
S
7
H
H
H
H
S
H
H
H
H
H
Note: For Hard Doubles, numbers indicate change from normal strategy, D becomes H
and H becomes D.
Hard Doubles
Normal Strategy
Dealer's Upcard
Player total
2
3
4
11
D
D
H
H
H
D
D
D
H
H
10
9
8
7
6
7
8
9
10
D
D
D
H
5
D
D
D
H
D
D
D
H
D
D
H
H
D
D
H
H
D
D
H
H
D
H
H
H
H
H
H
H
H
H
H
A
D
H
H
H
H
8
D
D
H
9
10
A
D
D
D
-1
-4
-4
H
H
H
H
H
H
H
H
H
H
8
D
13
9
10
A
12
H
H
H
H
H
6
H
H
H
H
Light Core Strategy
Dealer's Upcard
Player total
2
11
10
D
D
9
-1
8
7
4
5
6
7
D
D
D
-8
D
D
D
-8
D
D
D
D
D
-13
3
D
D
D
-8
H
H
H
H
H
H
-12
H
H
H
Heavy Core Strategy
Dealer's Upcard
Player total
2
11
D
10
12
9
H
H
H
8
7
4
H
H
5
D
D
8
H
H
H
H
3
D
12
2
D
15
4
58
6
D
D
16
H
H
7
D
D
H
H
H
4
H
H
1
H
H
H
-
Note: For Soft Doubles hands Ace-Six and lower indicate changes from D to Hand H to
D. Hands Ace-Seven and higher indicate changes from S to D and D to S.
Soft Doubles
Normal Strategy
Dealer's Upcard
Player total
Ace-9
Ace-8
Ace-7
Ace-6
Ace-5
Ace-4
Ace-3
Ace-2
2
3
4
5
6
7
8
9
10
A
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
H
H
H
H
D
D
D
H
H
D
D
D
D
D
D
D
D
D
D
S
S
S
S
S
D
D
H
H
H
S
S
S
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
D
H
H
H
H
H
8
9
10
A
S
S
S
S
S
S
S
S
-12
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
Light Core Strategy
Player total
Ace-9
Ace-8
Ace-7
Ace-6
Ace-5
Ace-4
Ace-3
Ace-2
2
-12
-9
-1
-1
-7
-12
H
3
-11
-6
D
D
-2
-11
-14
4
-12
-7
D
D
D
-5
-15
5
-13
-4
D
D
D
D
-12
H
H
H
H
Dealer's Upcard
7
6
-14
S
-3
S
D
-12
-7
D
D
H
D
H
-8
H
-4
H
Heavy Core Strategy
Dealer's Upcard
Player total
Ace-9
Ace-8
Ace-7
Ace-6
Ace-5
Ace-4
Ace-3
Ace-2
2
3
4
5
6
7
8
9
10
A
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
H
H
H
H
3
3
2
H
H
6
8
10
D
D
7
14
D
D
D
S
S
S
S
S
2
2
H
H
H
S
S
S
H
H
H
H
H
H
H
H
10
H
H
H
H
8
H
H
H
H
H
H
H
H
H
H
H
H
H
D
H
H
H
H
H
59
Note: For Splitting, numbers indicate changes from normal strategy, Y becomes Nand N
becomes Y.
Splitting
Normal Strategy
Dealer's Upcard
Player Cards
Ace,Ace
9, 9
8, 8
7, 7
6,6
4,4
3,3
2,2
2
Y
Y
Y
Y
N
N
N
N
3
Y
Y
Y
Y
Y
N
N
Y
4
Y
Y
Y
N
Y
N
Y
Y
8
Y
Y
Y
9
Y
Y
Y
10
Y
A
N
N
Y
Y
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
8
Y
Y
Y
-7
9
Y
Y
Y
10
Y
A
N
N
N
N
N
N
N
N
N
N
N
N
N
N
9
12
7
Y
10
10
A
N
N
Y
8
Y
6
Y
Y
Y
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
Y
Y
4
N
8
12
11
10
9
7
7
Y
5
Y
Y
Y
6
Y
Y
Y
Y
N
N
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
N
Y
Light Core Strategy
Dealer's Upcard
Player Cards
Ace,Ace
9,9
8,8
7, 7
6, 6
4,4
3,3
2,2
2
Y
Y
Y
Y
-1
N
N
N
3
Y
Y
Y
Y
Y
N
N
Y
5
Y
Y
Y
6
Y
Y
Y
7
Y
-6
Y
N
-6
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
-6
Y
4
Y
Y
Y
N
Y
-3
N
Y
Y
-7
Y
-12
N
N
N
N
Heavy Core Strategy
Dealer's Upcard
Player Cards
Ace,Ace
9,9
8,8
7,7
6, 6
4,4
3, 3
2,2
2
Y
Y
3
3
Y
6
Y
6
1
N
N
N
N
6
Y
5
Y
7
Y
6
Y
8
Y
N
N
N
N
N
N
N
3
N
2
Y
2
Y
2
Y
3
4
Y
60
15
Y
Y
7
Y
N
7
Works Cited
Blackjack - Odds and Strategies. April 11,2002. <http://www.thewizardofodds.comf>.
Casino Center. 11 April 2002. <http://www.casinocenter.comf>.
Casino Start-up. 11 April 2002. <http://www.casinogenerator.comf>.
Counting Cards. 4 April 2002. <http://www.freeblackjacktips.comlcountingcards.htmi>.
Frank, Scott. Blackjack for Winners: The Core System that Beats the Dealer. (Fort Lee
N.J: Barricade Books Inc, 1993).
Online Blackjack Strategy. 4 April 2002.
<http://www.blackjack-strategy.netlonline-blackjack-report.htm>.
Sifakis, Carl. The Encyclopedia of Gambling. (New York NY: Facts on File Inc, 1990).
Smith, Ken. Blackjack Info. 4 April 2002. <http://www.blackjackinfo.com>.
Tamburin, Henry. Double or Nothing, GR. 4 April 2002
<http://www.rgtonline.comlgamespage/artlisting.cfrnl1483>.
Thorp. Edward O. Beat the Dealer. (New York NY: Vintage Books, 1966).
Vogel, Phillip 1. Blackjack: The Real Deal. (Califon N.J: Raven House Publishing,
1997).
61