Sunday, December 14, 2014

Poker with Tarot Cards

So recently I was learning to play Poker along with an inquisitive 10-year-old named Eli, and the question came up: how would you play Poker with Tarot cards?

(The comedian Steven Wright is known for saying, “Last night I stayed up late playing Poker with Tarot cards. I got a full house and four people died.”—but in fact, the Tarot deck has been used for playing games since it appeared in Europe in the 15th century, presumably with few fatalities. According to the Wikipedia article on Tarot cards, the earliest record we have for the practice we now consider the primary purpose of the Tarot—telling fortunes, where each card has a particular significance—dates from the 18th century. So if historical precedence is any guarantee, you are most likely not courting spiritual disaster by sneaking in a few hands of Tarot Klondike.)

Of course, making the rules is always more fun than playing by the rules, so I'm hardly the first person to suggest alternate ways of playing Poker. Variations include:

The knobs that can be twiddled in designing a new Poker variant (ignoring the betting, which is a whole nother can of worms) include:

  • The deck:
    • The number of suits
    • The number of ranks
    • Whether all suits have the same ranks (is the deck a rectangular grid?)
  • The number of cards that count as a hand (could be more or less than the usual 5)
  • The types of card combinations that are scored:
    • Straights (card ranks are sequential)
    • Flushes (all cards are of the same suit)
    • n of a kind (that is, groups of n cards of the same rank)
    • Or something else, like the super mentioned above. You could also eliminate or modify one of the standard combinations—maybe flushes don't count for anything, or you allow a “small straight” (all but one of the cards' ranks are sequential)

Let the twiddling begin!

The deck of interest here is the 78-card Tarot deck of 5 suits, where the trumps have 22 cards and the other suits (the standard 4 from the 52-card deck) 14 cards each. The trumps are numbered 0 through 21; the other suits 1 through 10 and then Jack, Knight, Queen, King (or Valet, Cavalier, Dame, Roi in the French Tarot deck I have).

A 5-card hand seems a bit small for a deck that is half again as large as the standard one, so I'd prefer 6. Eli agrees, which settles it.

How to score the hands is a bit of a puzzle. What is a straight? Do we just do a sequential mapping of the trump ranks to the ranks in other suits, and decree that a Jack is an 11, a Knight a 12, a Queen a 13, and a King a 14? Somehow that doesn't seem very satisfactory. I gnawed on this problem for a while and finally decided that straights are based on the nominal ranks of the cards, and the permissible straights are defined by how those ranks are ordered in each suit. That is, the ranks in a straight must be a subsequence of either:

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 0


    1 2 3 4 5 6 7 8 9 10 J N Q K 1

(where the lowest card in each suit may also be the highest, for purposes of a straight, as is conventional).

For example, a 6-card straight 3-4-5-6-7-8 may consist of a mix of trump and nontrump suits, but a straight that begins with 8-9-10 must continue with either the 11-12-13 of trumps or the J-N-Q of some other suit(s), so as to preserve a sequence of ranks that actually appears in some suit.

Increasing the number of suits means that flushes become rarer, and increasing the number of cards in the hand means that straights become rarer. These both seem like aesthetic flaws, but I've decided to allow small straights (that is, sequences of 5 cards within a 6-card hand; big straights of 6 cards are also allowed), while ignoring the flush problem (you still need all 6 cards to be of the same suit for a flush. Mixing small straights and small flushes just makes the names of the hands too long).

Here's the ranking of hands (with frequency), based on the above choices:

     big straight flush 0.000023%
     small straight flush 0.000235%
     5 of a kind 0.000284%
     4 of a kind + pair 0.002403%
     3 of a kind + 3 of a kind 0.002412%
     flush 0.033468%
     big straight 0.049992%
     4 of a kind 0.052961%
     pair + pair + pair 0.097513%
     small straight pair 0.106750%
     3 of a kind + pair 0.353574%
     small straight 0.504444%
     3 of a kind 2.466884%
     pair + pair 6.093654%
     pair 40.184611%
     high card 50.050791%

A “small straight flush” consists of 6 cards of the same suit, but only 5 of them are in sequence. A “small straight pair” has 5 cards in sequence, and the rank of one member of the sequence is repeated in the sixth card (as in 4-5-6-7-8-5).

If the whole big straight/small straight thing gives you the willies, an alternative ranking that uses only 6-card straights is:

     straight flush 0.000023%
     5 of a kind 0.000284%
     4 of a kind + pair 0.002403%
     3 of a kind + 3 of a kind 0.002412%
     flush 0.033703%
     straight 0.049992%
     4 of a kind 0.052961%
     pair + pair + pair 0.097513%
     3 of a kind + pair 0.353574%
     3 of a kind 2.466884%
     pair + pair 6.093654%
     pair 40.291361%
     high card 50.555235%

And in case anyone is curious what happens in a standard 52-card deck with 5-card hands where you allow small (4-card) straights, the rankings are:

     big straight flush 0.001539%
     small straight flush 0.012159%
     4 of a kind 0.024010%
     3 of a kind + pair 0.144058%
     flush 0.184381%
     big straight 0.392465%
     small straight pair 0.650106%
     3 of a kind 2.112845%
     small straight 3.100471%
     pair + pair 4.753902%
     pair 41.606797%
     high card 47.017268%

For reference, the standard scoring in the standard deck is:

     straight flush 0.001539%
     4 of a kind 0.024010%
     3 of a kind + pair (full house) 0.144058%
     flush 0.196540%
     straight 0.392465%
     3 of a kind 2.112845%
     pair + pair 4.753902%
     pair 42.256903%
     high card 50.117739%

If you're interested only in standard scoring of 5-card hands in alternate decks, Travis Hydzik has a calculator that quickly figures out hand values for any deck with at least 4 suits and at least 5 ranks, where every suit has the same ranks. If you play with this tool a little bit, you'll see that 13 ranks is something of a sweet spot for decks with a moderate number of suits (between 5 and 9)—with fewer ranks, pairs become even more common than hands with nothing in particular (that is, hands ordered by high card), which strikes me as perverse.

So, I've answered the Tarot Poker question to my own satisfaction. If you want to play Poker with Tarot cards and make different rules, by all means go ahead. Discovering the analogies that work for you between the standard deck and a different deck is more than half the fun.

I thought I would be done by now, but I'm not. Traipsing around the Web in search of alternate card decks led me to something wonderful that you can actually buy: a Fanucci deck. If you are sufficiently Old School, you might recognize this as belonging to the Zork subgame Double Fanucci. What's wonderful about the Fanucci deck? Well, it's got 15 suits named Books, Bugs, Ears, Faces, Fromps, Hives, Inkblots, Lamps, Mazes, Plungers, Rain, Scythes, Time, Tops, and Zurfs—11 cards each, numbered 0 through 9, then ∞—along with 9 unranked trumps called Beauty, Death, Granola, Grue, Jester, Light, Lobster, Snail, and Time (that's right, there's a Time trump that is not part of the Time suit).

Thank you, Internet!

If you don't immediately want to play with something like that, then I just don't know what's wrong with you. Unfortunately, this is just a blog, so I can't actually play a game with the Fanucci cards with you here, but I can tell you what happens when you try to apply the standard Poker scoring rules to the Fanucci deck (5-card hands—I'm treating only the cards ranked 0 through 9 as able to participate in straights; the trumps have no numeric rank and so cannot participate in straights or n of a kind):

     straight flush 0.000007%
     flush 0.000555%
     5 of a kind 0.002634%
     4 of a kind 0.190342%
     straight 0.363255%
     3 of a kind + pair 0.418992%
     3 of a kind 4.593350%
     pair + pair 6.961714%
     high card 41.277900%
     pair 46.191251%

Hmm, high card is more common than a pair. A bit disappointing, but not unexpected, given how many more suits there are than ranks.

One way to try to fix Fanucci Poker is by not letting pairs count for anything, which gets you:

     straight flush 0.000007%
     flush 0.000555%
     5 of a kind 0.002634%
     4 of a kind 0.190342%
     straight 0.363255%
     3 of a kind 5.012342%
     high card 94.430865%

but this means that 94% of hands are scored by high card, which is way off the 50% for standard Poker. That can't be right.

More in the spirit of things is probably to reverse the role of ranks and suits: ranks count only when all 5 of your cards are the same (a flush-like 5 of a kind), whereas any 2 or more of the same suit count (an n-of-a-kind-like flush). So, for example, a straight 3-flush is a straight where 3 of the cards have the same suit:

     straight 5-flush 0.000007%
     straight 4-flush 0.000502%
     5-flush 0.000555%
     straight 3-flush + 2-flush 0.001005%
     5 of a kind 0.002634%
     straight 3-flush 0.013059%
     straight 2-flush + 2-flush 0.019589%
     4-flush 0.065484%
     straight 2-flush 0.156714%
     3-flush + 2-flush 0.163567%
     straight 0.172385%
     3-flush 2.518297%
     2-flush + 2-flush 4.194274%
     2-flush 39.967163%
     high card 52.724764%

This treatment makes 53% of hands scored by high card, which is satisfyingly close to conventional.

I've calculated the odds above with the help of a Scala program I wrote. Because I didn't trust myself to do the combinatorial math for quirky decks where the suits don't all have the same length, the program actually enumerates all possible hands (ignoring order) and checks each of them—this takes many hours on my laptop for 6-card Tarot hands or 5-card Fanucci hands. You're welcome to try the program out for yourself, but if you want to play 7-card Fanucci, you might want to take a different approach.

Anyway, I hope all this encourages you to get some unfamiliar cards and try something new. All decks on hand!

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