# Quantum Blackjack Analysis

## Introduction

Quantum Blackjack is a live dealer blackjack variant by Playtech. The game is dealt to a single player hand, which an unlimited number of players may bet on. If players don't all take the same action, then only those cards pertinent to each players hand will be shown. The twist is the game will randomly deal 1 to 3 cards from a separate deck. A multiplier is associated with each card. If the player gets a card to match a multiplier card, then he gets that multiplier if he wins. The take-back is if the dealer busts with three cards, all hands left standing push.

## Rules

1. The game follows conventional blackjack rules, which I assume the reader is familiar with, unless otherwise stated below.
2. An unlimited number of players may play on one single player hand. Each player may control is own fate. If some players stand, any further cards dealt to other players will not be shown to those players who stood.
3. Eight decks are used.
4. Dealer stands on soft 17.
5. The dealer take a hole card and peeks at it for blackjack.
6. In addition, one to three cards will be dealt from a separate single deck. These will be displayed on the screen. I will call these the Multiplier Cards.
7. Each Multiplier Card shall have a multiplier associated with it, from 3x to 10x. I do not know the probability distribution of each multiplier.
8. If at any point a card is dealt to the player that matches one of the Multiplier Cards, then any player will shall get that multiplier.
9. If the player earns more than one multiplier, the players actual multiplier will be the product of all earned multipliers. For example, if the player gets a 3x and a 5x multiplier, then his overall multiplier will be 15x.
10. Wins with a multiplier are expressed on a "for one" basis. For example, if the player bet \$100 and won with a 3x multiplier, his actual winnings would be \$200, plus the return of his \$100 wager.
11. The same Multiplier Card may not be used more than once. For example, if a Multiplier Card was the 7 of diamonds with a 10x multiplier and the player had three 7 of diamonds in his hand and won, then his overall multiplier would still be only 10x.
12. I assume the following other rules, but am not sure: no surrender, player may double on any two cards, player may double after a split, player may split once only (no re-splitting).
13. At Playtech, the 21+3 and Perfect Pairs side bets are offered. Multipliers do not apply to side bets. The side bet pay tables are below.

## Strategy

It isn't often that I say this, but working out a basic strategy for this game would be very difficult. It would be easy to say that the player is strongly incentivized to get as many cards on the table as he can, without taking ridiculous risks. In other words, the player should hit and split more, double and stand less. I'm afraid I can't be more specific than that.

## Analysis

All I can say about the main game is Playtech claims the return is 99.47%. I am quite sure that is based on optimal player strategy, whatever that is.

I can offer the following table that shows the probability of the dealer busting with three cards by dealer up card.

### Dealer Bust with Three Cards

Up
Card
Bust with 3 Cards
Probability
Ace 0.00%
2 9.46%
3 14.23%
4 19.58%
5 25.54%
6 31.92%
7 17.69%
8 17.64%
9 17.79%
10 17.77%
Average 17.30%

I can at least say that the 21+3 side bet follows the 5-10-30-40-100 pay table. The following return table shows a house edge of 3.70%.

### 21+3 — 8 Decks

Event Pays Combinations Probability Return
Suited three of a kind 100 2,912 0.000244 0.024446
Straight flush 40 24,576 0.002063 0.082524
Three of a kind 30 61,568 0.005169 0.155055
Straight 10 368,640 0.030947 0.309465
Flush 5 700,928 0.058841 0.294207
Loser -1 10,753,536 0.902736 -0.902736
Total   11,912,160 1.000000 -0.037039

The Perfect Pairs side bet follows the 6-12-25 pay table. The table below shows a house edge of 4.10%.

### Perfect Pairs — 8 decks

Hand Pays Combinations Probability Return
Perfect pair 25 1456 0.016867 0.421687
Colored pair 12 1664 0.019277 0.231325
Red/black pair 6 3328 0.038554 0.231325
Non-pair -1 79872 0.925301 -0.925301
Total   86320 1 -0.040964

## More Analysis

As long as a wrote a custom program for this game, here are the probabilities of various dealer outcomes by up card. This assumes the dealer always plays out his hand.

### Dealer Events — Stand on Soft 17 — 8 decks

Event Ace 2 3 4 5 6 7 8 9 10 Average
Dealer blackjack 0.308434 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.077108 0.047451
Total of 17 0.130210 0.139694 0.134482 0.130539 0.121946 0.165640 0.369048 0.128846 0.120231 0.111791 0.145215
Total of 18 0.130813 0.134518 0.130524 0.124534 0.122390 0.106208 0.137898 0.359800 0.118011 0.111608 0.139318
Total of 19 0.130643 0.129929 0.125313 0.121300 0.117607 0.106390 0.078478 0.128684 0.351581 0.111814 0.133629
Total of 20 0.130879 0.124021 0.120695 0.116451 0.112134 0.101592 0.078668 0.069263 0.120275 0.340560 0.179709
Total of 21 0.053595 0.118308 0.114871 0.111707 0.108021 0.097247 0.073878 0.069452 0.060864 0.034738 0.072838
Bust with 3 cards 0.000000 0.094616 0.142297 0.195751 0.255352 0.319236 0.176940 0.176393 0.177871 0.177685 0.173015
Bust with 4 cards 0.077930 0.186456 0.177740 0.161866 0.136237 0.086546 0.073111 0.059515 0.045791 0.031980 0.087163
Bust with 5 cards 0.032068 0.062196 0.047464 0.033718 0.023657 0.015599 0.011055 0.007532 0.005068 0.002614 0.019139
Bust with 6 cards 0.004990 0.009388 0.006106 0.003847 0.002499 0.001462 0.000885 0.000498 0.000298 0.000101 0.002337
Bust with 7 cards 0.000415 0.000826 0.000484 0.000275 0.000151 0.000077 0.000039 0.000016 0.000009 0.000001 0.000177
Bust with 8 cards 0.000022 0.000046 0.000024 0.000011 0.000005 0.000002 0.000001 0.000000 0.000000 0.000000 0.000009
Bust with 9 cards 0.000001 0.000002 0.000001 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000