The “21 + 3” blackjack side bet is based on examining the player’s two cards and the dealer’s up-card. If the three cards form a flush, straight, three-of-a-kind or straight flush, the player wins. In the original version, the payout for each of these was 9-to-1. With this pay table, the game has a house edge of 3.2386%. Recently, new pay tables have been introduced that have higher house edges and greater volatility.

The point of attack I considered is to target flushes. Any strong imbalance in the suits favors the player. For example, consider a situation where there are 40 cards, 10 of each suit. Without going into the math, the number of ways of making a three-card flush is 480. Now, take those same 40 cards, and assume they are distributed 15, 10, 10, 5. Then the number of three-card flushes is 705. The more unbalanced the distribution of suits, the more the edge swings towards the player.

To make use of this, it is necessary to keep track of the number of cards in each suit that remain in the shoe. This can be accomplished by a team of counters, each keeping track of one of the suits (or by a mentally gifted solo counter). The counters then compute the difference between the most abundant and least abundant suits. This difference is then turned into a true count, and if that true count is sufficiently large, the player has an edge.

I created a simulation to model using this system on a six-deck shoe game dealt to 52 cards and simulated one hundred million (100,000,000) shoes. This work showed that a counter can gain an edge on approximately 3.5% of the hands dealt (1.75 hands per shoe). The counter should make the 21+3 wager whenever the true count is 8 or higher. The average edge when the wager is made will be just over 5%. If the table limit is $25, then a counter playing heads-up can earn about $2.20 per shoe. The new pay tables were not evaluated.

As an experiment, I shuffled one hundred thousand (100,000) shoes and computed the edge at the point when there were 100 cards remaining in the shoe. The result of this simulation was an average house edge of 3.247%, which is close to the theoretical value of 3.239%. More interesting was that the standard deviation of the house edge was 3.57%. It follows that a player edge is 0.910 standard deviations above the mean. Therefore, the player will have an edge on about 18.14% of the shoes at that point. The trick is knowing which ones. My simulation gave a maximum player edge of 23.71% and a maximum house edge of 13.55%.

There are two reasons that APs will not target 21+3 with this system. The first is its complexity, the second is the low return. However, there is another approach that may be significantly stronger.Consider a shuffle tracking approach where a slug of cards is identified that is either deficient in one suit or abundant in one suit. In this case, by tracking that slug through a weak shuffle, the AP will have a good opportunity. My knowledge of shuffle tracking is minimal. I cannot say if this is an approach that has been used in practice. Finally, I have not considered if the new pay tables have a similar vulnerability to the 9-to-1 pay table.

For more information on this topic see:

The following are my recommendations regarding 21+3:

  • Watch for solo players who play this wager only at certain times during the shoe, and when they play, it is a maximum bet. These players may be using a shuffle tracking approach.
  • Watch for a team of players who all play maximum wagers on this bet at the same time. These players may be using a card counting approach.
  • If your game is hand shuffled and if that shuffle is potentially trackable, consider a new shuffle or leasing an automatic shuffler.
About the Author
By

received his Ph.D. in Mathematics from the University of Arizona in 1983. Eliot has been a Professor of both Mathematics and Computer Science. Eliot retired from academia in 2009. Eliot Jacobson