Every figure in the book is produced by enumerating the game completely - all 22,100 three-card hands against every hand the dealer can hold behind them. Nothing is sampled and nothing is repeated from anybody else. This page is about how that is done and what falls out of it; the figures themselves, and the reasoning, are in the book.
Twelve chapters. Nine of them turn on a figure that had to be computed rather than looked up. Here is the shape of each - the headline number, and what had to be counted to get it. The reasoning, the full tables and the rest of the figures are in the book.
Three Card Poker resolves in about forty hands an hour, which is roughly twice the rate of blackjack. Speed is the mechanism: the decisions are small, they come quickly, and a small mistake made forty times an hour stops being small.
Every one of the 22,100 hands you can be dealt is ranked by what playing it is worth against the flat −1.000000 that folding always costs. The line falls where the list crosses, and it crosses between two hands that differ by one card. The cover of this book is the hand it crosses at.
Four Pair Plus pay tables in common use, each priced from the full distribution. The spread between the best and the worst is wider than most players would guess, and the two best differ by a single digit in a single place - for the hand you make most often, which is why it matters so much.
The one thing on the layout paid to you whether the hand wins or loses. Broken down by hand, it turns out the impressive payouts on the sign contribute almost nothing and the humblest hand on the list carries nearly two thirds of the value.
The bet most players make without looking. Whether it is reasonable depends entirely on the sign above it - on one table it is among the fairer side bets in the casino, and on another it is one of the worst.
Your three cards and the dealer's, played as one five-card hand out of six. All 407,170,400 splits counted. The royal flush that sells the bet is rarer than most people believe by an order of magnitude - and the progressive beside it is the only wager in the game whose value changes while you are sitting there.
The California game turns one of the dealer's cards face up before you act. The memorised rule stops applying immediately: the correct decision is re-solved against the 1,128 hands the dealer can still finish with, and the line moves a long way depending on what is showing.
Some rooms charge you by the hand rather than in the pay table. It is taken before a card is dealt, it does not care what you bet, and it cannot be won back. Expressed against the bet rather than the hand, it produces the one genuinely counter-intuitive instruction in the book.
Rate of play, times average bet, times the blended edge across everything you have out. Three numbers. It is the figure the floor already has about you and almost no player has about themselves - and it is not frightening, just real.
What to sit down with, what a table actually wants in front of you, and why the standard advice about buy-ins is built on the variance of the game rather than on anybody's superstition.
The floor values you on theoretical loss, not on whether you are winning. Knowing the formula tells you what you are worth to them, what that buys, and what it costs to be worth it - which is usually more than the comp is.
Drilling the decision against the exact answer until it is automatic. The Simulator on this site does the same enumeration in your browser, marks every call you make, and totals the expected value your mistakes have given away.
Every figure in the book was produced by counting - completely, from the deck outward - rather than by simulation or by repeating a number from somewhere else. The game is small enough that there is no reason to do anything else.
| Step | Size | What it settles |
|---|---|---|
| Hands you can be dealt | 22,100 | The complete distribution |
| Dealer hands behind each | 18,424 | Every way a hand can be beaten |
| Confrontations walked | 32,334,120 | 1,755 distinct hands x 18,424 dealer hands. The full space is 407,170,400 pairings; suit isomorphism reduces the work, not the answer. |
| Six-card splits | 407,170,400 | The entire 6 Card Bonus |
| Margin of error | ±0.000000 | Counting leaves nothing to estimate |
Most published figures for casino games come from simulation - deal a billion hands, measure what happened, quote the average. That is a sound method and it leaves a confidence interval: run it again and the third decimal moves. Three cards from fifty-two is 22,100 hands, which is small enough to count all of instead. Counting does not produce an estimate to be checked; it produces the answer, and running it again produces the same one.
Every player hand against every dealer hand that can follow it, and every six-card deal split twenty ways between you and the dealer, are the same number of things. So a single pass produces both the play/fold solution and the whole six-card distribution at once - and guarantees they agree, because they came from the same count.
Three Card Poker was invented, and it was patented. Derek Webb filed in July 1994 and the patents that followed ran their twenty years and expired. The method of playing the game has been in the public domain for a decade. What is still owned is the name.
| Patent | What it covered | Status |
|---|---|---|
| US 5,685,774 | The game itself | Expired 19 July 2015 |
| US 6,237,916 | The game with the Pair Plus wager, including video and online play | Expired July 2015 |
| US 7,264,243 | The six-card version | Expired 19 April 2016 |
| US 2008/0111308 | The dollar progressive, paying a share of the meter for a mini royal | Abandoned 2009, never granted |
Nothing similar could be filed today. In Alice in 2014 and In re Smith in 2016 the courts put the rules of a wagering game outside what a patent can cover: rules are an abstract idea, a standard deck supplies no inventive concept, and shuffling and dealing are, in the Federal Circuit's words, conventional and traditional practices. Casino games are protected now by what surrounds them - the linked meters, the shufflers, the licensing - rather than by the play itself.
A trademark is a different thing and does not expire while it is used. Three Card Poker® is a registered trademark of Light & Wonder, Inc., and this book and site are independent of them. The name identifies the game the book is about. The mathematics belongs to nobody.
And all of it is in The Simulator, computed live while you play, so you can check any of it yourself.
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