21 plus 3 Blackjack
One of the first games I posted about on this blog was the blackjack side bet Twenty One Plus Three (21+3). In this post, I showed that there was very little possible profit against 21+3, even using perfect play. Nevertheless, this post on card counting 21+3 is the most viewed blackjack side bet article in this blog, getting about 70% more hits than this second place post on Lucky Ladies. The 21+3 post ranks 6th in overall views. It continues to be one of the top posts accessed each day. I don’t know why.
For over two years, I have known about a version of 21+3 called “21+3 Extreme” (21+3E). This version pays for the same events as 21+3, but uses a pay table leading to a higher house edge and greater volatility. Because of the extremely low vulnerability of 21+3 to advantage play and the higher house edge of 21+3E, I long ago concluded that 21+3E would be a waste of time to analyze. However, given the continued interest in 21+3 (and a lack of anything better to do), I decided to finally give 21+3E a look. The results of this analysis met my extraordinarily low expectations.
This post covers the six-deck case. With fewer decks, the house edge for 21+3E increases, making it even less vulnerable.
The following tables give the combinatorial analysis for 21+3 and 21+3E:
In particular the house edge for 21+3 is 3.24%, whereas the house edge for 21+3E is 13.39%. The term “Extreme” certainly describes this side bet: it is “Extremely” good for the casino and “Extremely” bad for the player.
The following Excel spread sheet allows the user to input the exact composition of the shoe and the pay table for the variation of 21+3 in use; it outputs the house edge, hit frequency and standard deviation. I suppose in some universe a spread sheet like this could be used for advantage play against 21+3. Then again, in some universe, a Komodo dragon is the current world chess champion.
In investigating card counting 21+3E, the AP can do no better than perfect play (for example, by using the spread sheet given above), where the AP makes a wager on 21+3E if and only if he has the edge. The following table shows the results of perfect play against 21+3E in the six-deck case when the cut card is placed at 260 cards. These results confirmed my intuitive expectation that 21+3E has almost no card counting vulnerability:
The opportunity to make a 21+3E wager with the edge occurs about 3 times per 1000 hands. When the player makes a 21+3E wager, his average edge is about 4.08%. This leads to a gain of about 0.0122 units per 100 hands. If an AP is making a $100 wager whenever he has the edge over 21+3E, and otherwise does not make a wager, then he will earn about $1.22 per 100 hands.
The final number given in this table, the desirability index (DI), is discussed for blackjack side bets in this post. The value 0.73 given in this table shows that 21+3E has “minimal vulnerability” to advantage play. (I note that I used the standard deviation from the combinatorial analysis for 21+3E in computing the DI).
Gambit is a television game show, created by Wayne Cruseturner and produced by Heatter-Quigley Productions, that originally ran on CBS from September 4, 1972 to December 10, 1976. A slightly retooled version, Las Vegas Gambit, aired on NBC from October 27, 1980 to November 27, 1981, originating from the Tropicana Hotel and Casino in Las Vegas...
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What is 3x plus 6 equals 21?
3X + 6 = 21
subtract 6 from each side
3X = 15
divide both sides integers by 3
X = 5
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3(5) + 6 = 21
15 + 6 = 21
21 = 21
========checks