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The "Rule Of" Rule

For whatever reason, I've written about single-deck Blackjack quite a bit lately, but it's a fascinating game when you count cards - profitable, too! If you've read many of my articles on winning at Blackjack, you've undoubtedly noticed that I often dicuss the importance of "penetration", which is another way of saying how far the casino deals into the deck(s) before shuffling the cards. Given a game with lousy rules, but deep penetration and almost any card counter can beat it. On the other hand, a casino can have great rules, but if they deal out, say, only half the cards, it's a tough game to beat, believe me.

The reason deep penetration is so important to a counter is that we can identify extremely profitable situations and take advantage of them when they happen and, quite simply, the deeper the casino deals into the deck, the more of those opportunities we'll see. Yes, it works both ways, so deeper penetration also causes much lower counts where the casino has a bigger edge but you need to remember that when the count is in the casino's favor, we will usually be making a minimum bet, if we're betting at all.

Whenever you see me talking about penetration, it's invariably presented as a percentage of the total deck or decks, depending upon the game I'm talking about. For example, in a six-deck game, 75% penetration means that 4.5 of the 6 decks involved are dealt out before the shuffle. If a single-deck game offers 50% penetration, that means 26 cards are dealt out before the shuffle.

But there's another way to measure the penetration of a single-deck game that's usually called "the rule of" rule, thus the title of this article. The "rule of" rule was developed by the casinos to make it easy for their dealers to know when to shuffle. Typically, a multi-deck game dealt from a shoe has a shuffle card that is inserted at the desired penetration level and when it comes out, the dealer will usually finish the round in play, then shuffle. When using the "rule of" rule at a single-deck game, the dealer does a simple calculation at the beginning of the deal, then s/he will know when to stop for a shuffle. I can best explain this with some examples.

Let's say that XYZ casino's policy is to deal their single-deck game by "the rule of 5". This means that any combination of players and the number of rounds (one completed hand to each player is a "round") they're dealt before a shuffle should add up to 5. Thus, three players would get two rounds, then the dealer would shuffle; four players would get one round and then the dealer would shuffle and so forth, Another example: In a "rule of 7" casino, four players would get three rounds before the shuffle, three players would get four, five players would get two, etc. See how that works? The dealer just has to subtract the number of players from the "rule of" rule for his or her casino to get the number of rounds that should be dealt before the shuffle. Obviously, a player may leave or join the table and throw this caculation out of kilter a bit, but in my experience, the dealer will either cut down on the number of rounds in the case of someone joining the table (assuming that the casino allows "mid-shoe entry") or just stay with the plan if someone leaves. The dealer's hand is typically not included in the calculation as it's made by the casino, but we counters include it when we do our calculation of penetration.

That brings us to the nifty part of this. If a casino uses the "rule of" rule, you can accurately calculate the penetration of their game and that will tell you if the game is worth playing. An interesting statistic of Blackjack is that the average hand is composed of 2.7 cards. Thus, with four players, one round of play will use up 4 x 2.7 = 10.8 +2.7 = 13.5 cards, on average. You can see that 4 players use 10.8 cards (yes, I know you can't get 8/10s of a card; we're talking averages here) and the dealer uses 2.7, so the total for one round of play is 13.5 cards. If this is a rule of 7 casino, the four players will get three rounds before the shuffle. So, all we need to do is multiply 13.5 times 3 and we get 40.5 cards. That's almost 80% penetration! Sure, it will vary at times, particularly when that guy at first base hits his A,2 with two more Aces, a 2 and a 4, but for the most part, it'll work out to be 2.7 cards for each player and the dealer.

As you can see from the example above, three rounds to four players is a great situation for the counter. But, the rule of 7 also applies to five players and they would get two rounds under the formula. The math on that is: 5 x 2.7 = 13.5 + 2.7 = 16.2 x 2 = 32.4, which is 62% penetration. Still good, but it's not 80%. Plus, there's another problem with this scenario. A game of "two rounds and shuffle" is difficult to beat because you need the first round of play to determine the count in order to figure your bet, then you place it for the second round, so the first round of play (if you're smart) is always played at the minimum bet. What you're left with is a game where you're making a minimum bet on the first round and then are trying to shove out as many $$$ as possible on the second round. That's not good for keeping the welcome mat out at the casino of your choice. So, it's fair to say that there are good "rule of 7" situations and bad "rule of 7" situations. That's true for most of the "rule of" rules, I suppose. However, if the "rule of" number gets too low, then nothing will help.

Okay, what does all this mean to the counter, in terms of percent advantage? It's a fair question and I have some answers. While doing a "GameMaster Tutor" article on good single-deck games, I ran a ton of simulations on single-deck games using Statistical Blackjack Anayzer 5.0, which is a fabulous software program that allows you to input all kinds of rules, plus various "rules of" scenarios. By keeping all of the variables (the casino's rules, counting system, bet schedule, etc.) the same and varying only the "rule of" rule, we can get a good idea of what it's worth to us. One thing to remember is that a "rule of" rule can have various permutations, i.e,, four rounds to two players, three rounds to three players, two rounds to four players are all "rule of 6". As you'll soon see, not only do we want to determine the general "rule of" rule for a casino, we also want to play only when the table has an appropriate number of players, whenever possible. A good example of a bad "rule of 6" is one round to five players. Not only is the penetration shallow, but there's little point in playing because you'll always be betting the table minimum due to the fact that each round is coming from a freshly-shuffled deck. Here's a look at a rule of 6 game in its various pemutations, other than one round to five players:

Rule of 6 Simulations

#1 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 6: 2 rounds to 4 players. Player's edge: 0.80%, SCORE: 33.67

Comments: Because I always recommend you not play at less than a 1% overall edge, this is a game to pass up. "SCORE" stands for Standardized Comparison of Risk and Expectation. In his book, "Blackjack Attack" in the 2nd edition, Don Schlesinger explains a method of comparing games, which he calls SCORE and that takes into account different situations like betting only when the count is at +1 or more, or playing "all", etc. The program I used, Statistical Blackjack Analyzer, also calculates the SCORE whenever possible. What those numbers represent is the anticipated hourly win rate in dollars for a player who has a $10,000 bankroll, along with other conditions, such as a specified risk of ruin and other factors. I'll leave it to you to get the book if you'd like more information on SCORE, but suffice it to say that this is a valid means of comparison for various games. By the way, a game with a SCORE of under 50 probably isn't worth your time.

#2 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 6: 3 rounds to 3 players. Player's edge: 1.20% , SCORE: 74.30

Comments: This game is worth playing, for sure. If one of the other players leaves, try spreading to two hands to keep the "configuration".

#3 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 6: 4 rounds to 2 players. Player's edge: 1.27% , SCORE: 82.84

Comments: The difference in the player's edge for this and the 2 rounds to 4 players rule of 6 game is that there are fewer players to "eat up" the cards in high counts.

#4 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 6: 5 rounds to 1 player. Player's edge: 1.02% , SCORE: 53.96

Comments: This game is barely acceptable, but you can get a lot of hands/hour.

Rule of 7 Simulations

#5 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 7: 2 rounds to 5 players. Player's edge:1.02%, SCORE: 54.52

Comments: To get the edge shown, you have to jump your bet on the second round to whatever the true count calls for and that will probably get you 86'd from the casino.

#6 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 7: 3 rounds to 4 players. Player's edge: 1.66%, SCORE:141.58

Comments: Now here's a game I can get into! The problem is that rule of 7 games are rare.

#7 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 7: 4 rounds to 3 players. Player's edge: 1.95% , SCORE: 196.18

Comments: Whoa! Unfortunately, some - but not all - casino personnel know this is a great game. Obviously worth looking for, though.

#8 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 7: 5 rounds to 2 players. Player's edge: 1.80% , SCORE: 165.96

Comments: Can't you just see yourself and a pal playing this game for, say, about two weeks straight?

#9 Single deck, dealer hits soft 17, double on any first two cards, no double after split, 1-4 bet schedule (1 unit at a true count of 0 or lower; 2 units at a count of 1; 3 units at 3; 4 at 4 or more). Basic strategy variations are used. Rule of 7: 6 rounds to 1 player. Player's edge: 1.35% , SCORE: 93.12

Comments: Rock and roll! Not only will you get a great game, but it'll come at a rate of 200+ hands per hour. We should all be so lucky.

Using this information

A lot of the games listed above look great and, indeed they are. That's the good news. The bad news is that rule of 7 games are extremely rare and rule of 5 games are pretty common. Just how bad is a rule of 5 game? Well, if you're getting 3 rounds to 2 players, the player's edge is only 0.80% and the SCORE is 37.22, clearly a game to avoid. What you need to do when you can, is use playing a second hand to make a marginal game into a good one. For example, if you go to a second hand in a game where 2 players ( you and someone else) get just 3 rounds before the shuffle, put out a second hand and see what the dealer does about it. If s/he still deals 3 rounds, you've gone to a rule of 6 game (3 rounds to "3" players), which has a player's edge of 1.20% and a SCORE of 74.30, a nice increase, no matter how you measure it.

Another good example of the use of a second hand is where you're playing a rule of 6 game with 1 other player (4 rounds to 2 players). If the count goes up, spread to 2 hands and, if that doesn't cause the dealer to shuffle, you'll now be playing a rule of 6 game where 3 rounds are dealt to "3" players. There's no big loss in the player's edge (it drops to 1.20% from 1.27%), but you'll be "eating up" the good cards and will hopefully win more than the other guy until the shuffle comes.

Another nice thing about these sims (my, aren't I proud!) is that you'll know what to do if you're in a rule of 6 game with 3 other players (2 rounds to 4 players) and one of them leaves. Now you have a game of 3 to 3 and that's an excellent situation. In that case, do not go to a second hand, because you're now playing a much better game (1.20% player's edge vs. 0.80%).

But the best thing that these sims demonstrate is that you should never play when there are more than two other players, beside yourself. You can easily see that the best penetration occurs when there are three players or less. And don't forget to count yourself in that calculation. Also, avoid rule of 5 games like the plague, unless you can turn it into a rule of 6 game..