Card counting

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Card counting is a strategy that enables the skilled Blackjack player to make a long-term profit. Blackjack became the most popular casino game in the 1960s and 1970s after the publication of Beat the Dealer by Edward Thorp, which introduced manageable card counting systems.

The larger the proportion of aces, face cards, and tens remaining in the deck, the more favorable the odds are for the player. Players take advantage of this by increasing their bet sizes and altering playing strategy. Some casinos consider this cheating and will ban card counters from playing.

That the player can win with card counting results from a simple rule: Although the player can stand or hit on any total, the dealer must take a hit until reaching 17. The greater the richness of the deck in ten-value cards than a neutral deck, then, the greater the chance of the dealer busting, leading the player to place larger bets and make less conservative strategy choices in such cases. The converse is true is ten-value card poor decks.

Although blackjack dilettantes create an encyclopedia of analysis and manufacture unwieldly jargon to provide a scientific aura around card counting, that rule is the dominant mathematical factor and the basis for all card counting systems.

Due to the need for perfect basic strategy playing, close attention, and self-discipline, it is very rare for even highly trained players to make a profit at blackjack. With even the best systems maxing out at around a 1% player advantage, statistical irregularities may bankrupt players who have insufficient bankrolls to ride out the inevitable losing streaks. Casino countermeasures such as more frequent shuffling and larger numbers of decks have made favorable circumstances more rare.

Blackjack is best enjoyed for entertainment with no money involved or with the expectation that money will be lost. However blackjack card counting systems are fascinating for anyone interested in gaming or statistics.

Summary table[edit]

System Name 2 3 4 5 6 7 8 9 10 A Developer
Hi-Lo 1 1 1 1 1 0 0 0 -1 -1 Harvey Dubner, Edward O. Thorp
Hi-Opt I 0 1 1 1 1 0 0 0 -1 0 Charles Einstein, Lance Humble, Carl Cooper
Hi-Opt II 1 1 2 2 1 1 0 0 -2 0 Charles Einstein, Lance Humble, Carl Cooper
Red Seven 1 1 1 1 1 0/1 0 0 -1 -1 Arnold Snyder
Uston Strongest and Simplest (SS) 2 2 2 3 2 1 0 -1 -2 -2 Ken Uston
Wong Halves .5 1 1 1.5 1 .5 0 -.5 -1 -1 Stanford Wong
Zen 1 1 2 2 2 1 0 0 -2 -1 Arnold Snyder


Most systems assign a value to every card, often zero, +1, or -1. The player keeps a running count. If the running count rises to a certain threshold, the player increases bets or plays more aggressively.

True count = Running count / Decks remaining [1]

All 10, J, Q, and K are treated the same, represented here as "10"


The Hi-Lo system was developed by Harvey Dubner in 1963 and popularized by Edward Thorp.[2][3][4][5]

  • 2,3,4,5,6 = 1
  • 7,8,9 = 0
  • 10, A = -1

Hi-Opt I[edit]

Hi-Opt created by Charles Einstein in 1968. Hi-Opt I by Lance Humble and Carl Cooper.

  • 3, 4, 5, 6 = +1
  • A, 2, 7, 8, 9 = 0
  • 10 = -1

Hi-Opt II[edit]

Hi-Opt II [6]

  • 2,3 = +1
  • 4,5 = +2
  • 6,7 = +1
  • 8,9 = 0
  • 10 = -2
  • A = 0

Knock-out (KO)[edit]

Omega II[edit]

Red Seven[edit]

Red Seven or Red 7 was developed by Arnold Snyder. [7][8]

  • 2,3,4,5,6, Red 7 = +1
  • Black 7, 8, 9 = 0
  • 10, A = -1

Uston Strongest and Simplest (SS)[edit]

Strongest and Simplest (SS) or Uston SS was developed by Ken Uston.[9][10][11]

  • 2,3,4 = 2
  • 5 = 3
  • 6 = 2
  • 7 = 1
  • 8 = 0
  • 9 = -1
  • 10, A = -2

Wong Halves[edit]

Wong Halves, Stanford Wong, 1975.[12][13]

  • 2 = 0.5
  • 3,4 = 1
  • 5 = 1.5
  • 6 = 1
  • 7 = 0.5
  • 8 = 0
  • 9 = -0.5
  • 10 = -1


Zen Count, Arnold Snyder, 1983.[14]

  • 2,3 = 1
  • 4,5,6 = 2
  • 7 = 1
  • 8,9 = 0
  • 10 = -2
  • A = -1

The Golden Diagram[edit]

After the publication of Beat the Dealer by Thorp, gambling casinos reacted to the advantage that a card counter gains over the house by adopting counter strategies. These included employing multiple decks rather than the single hand-held deck. Two-deck games and games employing four and six decks dealt from a so-called shoe became commonplace.

Players soon realized intuitively that both these changes in the game reduced their probabilities of winning. In games with a multiple deck, compared to single-deck or double-deck games, players experience frequency, magnitude, and depth (the fraction of the deck which has been dealt in playing previous hands) effects: 1) The deck becomes favorable less frequently at all depths, 2) when the deck does becomes favorable, the magnitude of the advantage is not as great, 3) all decks are favorable infrequently until a significant portion of the deck has been dealt and this occurs at greater depths into the deck in games using multiple decks.

Blackjack professional and writer Les Golden, based on a Monte Carlo simulation and theoretical arguments, calculated the magnitude of these effects. The results of his analysis are displayed as Golden diagrams.[15][16] He also suggested a stepwise betting strategy to reduce the effects.[17][18]

The Golden Diagram was presented in 2016 as a paper to the 16th triennial International Conference of Gambling and Risk Taking in Las Vegas, sponsored by the International Gaming Institute of the University of Nevada, Las Vegas. That paper and the paper he presented on a physics-based system for roulette resulted in the International Gaming Institute, University of Nevada, Las Vegas, naming him a 2016 Scholar.

The details of the Golden Diagram calculation are presented in a sequence of chapters in Golden's Venusian Casino Junkets and Other Essays in Mathematics and the Probabilities of Gambling, scheduled for early 2022 publication by World Scientific Publishing Co.

Blackjack writers[edit]

This is a list of blackjack players and/or theoreticians whose works are published by reputable (non-vanity or self-publishing) publishers.

John Bukofsky[edit]

Bryce Carlson[edit]

Daniel Dravot[edit]

Greg Elder[edit]

Les Golden[edit]

Les Golden devised the Golden Diagram. It was published in The Mathematical Scientist, Bluff Europe, and the Proceedings of the 16th International Gambling and Risk Taking Conference (2016). Golden was named a 2016 Scholar by the International Gambling Institute (IGI) of the University of Nevada, Las Vegas, for his scholarly contributions.

Fred Renzey[edit]

Chris Statz[edit]

Edward Thorp[edit]

A pioneering book codifying the methodology by Edward Thorp was published in 1962: Beat the Dealer: A Winning Strategy for the Game of Twenty-One.[19]

Julian Braun[edit]

Programmer whose work Thorp and others relied upon. Few copies of his book, now out-of-print, are available.

Nathaniel Tilton[edit]

Olaf Vancura and Ken Fuchs[edit]

Authors of Knock Out Blackjack.

Stanford Wong[edit]

Author of the books Basic Blackjack, Professional Blackjack, and Blackjack Secrets.

Frank Scoblete[edit]

Author of Golden Touch Blackjack Revolution.

Arnold Snyder[edit]

Author of books Blackbelt in Blackjack, Big Book of Blackjack, and Blackjack Wisdom..

Don Schlesinger[edit]

Author of Blackjack Attack.

Barry Meadow[edit]

Author of Blackjack Autumn

Rick Blaine[edit]

Author of Blackjack Blueprint

Ian Andersen[edit]

Author of Burning the Tables in Las Vegas and Turning the Tables in Las Vegas.

Maverick Sharp[edit]

Author of the 2013 Dynamic Blackjack.

Ken Uston[edit]

Author of Ken Uston on Blackjack and Million Dollar Blackjack..

Lawrence Revere[edit]

Author of Playing Blackjack as a Business.

Peter Griffin[edit]

Author of The Theory of Blackjack.

Lance Humble and Carl Cooper[edit]

Authors of The World's Greatest Blacklack Book.

In the Media[edit]

Never Split Tens!, a novel based on the life of pioneering blackjack card counting theorist Edward O. Thorp, by gambling writer Les Golden of Oak Park, Illinois, was published in 2017 by Springer. It includes numerous examples of card counting using Thorp’s systems in simulated casino play including changing bet sizes, group card counting, camouflage techniques, and entering play when the deck is favorable to the player.

See also[edit]

In the Media[edit]

Never Split Tens!, a novel based on the life of pioneering card counting theorist Edward O. Thorp, by Les Golden published in 2017 by Springer, includes numerous examples of card counting in simulated casino play including changing bet sizes, the Kelly criterion, group card counting, and entering the play when the deck is favorable to the player.


  15. Golden, Les (2010). “Countering the Casino Countering of Counters: The Golden Diagram to the Rescue,” Bluff Europe, June, p. 84-85
  16. Golden, Les (2011). “Trust Me: An Undetectable Winning System For Blackjack! ,” Bluff Europe, March, p. 94-95
  17. Golden, Leslie M. (2011). “An Analysis of the Disadvantage to Players of Multiple Decks in the Game of 21.” The Mathematical Scientist, 32, 2, p. 57-69
  18. Golden, Les (2011). “Stepping Out With My Baby: The Stepwise Betting Strategy,” Bluff Europe, April, p. 92-93
  19. Thorp, E. O. (1962, 1966) Beat the Dealer: A Winning Strategy for the Game of Twenty-One, Random House, New York