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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 isR$25, 🧾 then a counter playing heads-up can earn aboutR$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:

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