Does Mathematics Count?

Several weeks ago I was talking with Bib Ladder, a colorful Vancouver area poker player, at one of the local cardrooms about a home game he regularly inhabited twenty years ago.

``Well, ya see, professor, it was my first introduction to 3-card monte. Several of the players regularly dealt it. As time went on I observed that three-card flushes seemed to occur more frequently than three-card straights, but they played that a flush beats a straight. I didn't know what to think, so I asked a friend of mine what he thought. He was rather surprised and told me without blinking an eye that a straight beats a flush. Well, them old geezers in the game would have none of my contention that they were doin' it wrong so I just gave up tryin' to set things straight. I've often wondered about it and was thinkin' you might explain it to me even though I'm not much of a numbers man.''

``Bib, let's go sit at one of the tables and I'll give you a first lesson in counting,'' thinking to myself as I said this that there are probably more than a few poker players like Bib who don't really understand why hands are ranked as they are. Retrieving my ubiquitous pad of paper, I continued, ``Bib, I presume it is clear that when we count the number of types of hands, the fewer there are of a given type, the higher that type of hand is ranked.''

``I know that,'' he replied.

``Counting objects is a branch of mathematics that students find difficult because there is no universal machine that answers all problems. The methods are somewhat ad hoc and to some people seem magical and without form. There is almost an art to solving counting problems. Still, there are several basic principles to keep in mind. Primarily, you have to know when to add and when to multiply. A rule-of-thumb that works reasonably well is to multiply when the word ``and'' is involved, and to add when the word ``or'' is involved. Finally, you must use common sense. All of these points are illustrated nicely in counting three-card poker hands, but first let's consider your particular question that deals with straights and flushes.

Let's count three-card straight flushes first. We'll see why we need to do this in a few minutes. So, Bib, is there some unique card that completely determines any particular three-card flush?''

``Sure,'' he replied. ``Once you know the smallest card in the three-card straight flush, you know everything.''

``Right! You have discovered a unique marker that determines the objects being counted. In this case, the smallest card. We could have used the middle card or largest card as well. Now all we have to do is count the possible number of smallest cards. A king is the only card that cannot be a smallest card in a three-card straight flush so that there are 48 possible smallest cards. Thus, there are 48 three-card straight flushes.

We now move to three-card straights which have the form x,x+1,x+2. How many cards can appear as x?''

``Well, it can be any of 48 cards since a king is the only card that cannot begin a three-card straight,'' Bib answered.

``Right! And once x has been determined, x+1 can be any of 4 cards as can x+2. We multiply because we are choosing one card and a next and a next, that is, the word `and' is involved. Multiplication yields 768 straights. Now is when we use common sense by realizing that three-card straight flushes have been included in the latter count so that they must be removed. Therefore, there are 720 straights. Note that there are 15 times as many straights as there are straight flushes. This lets you know just how much stronger a straight flush really is.

Let's consider three-card flushes next and see if you can do it. How many three-card heart flushes are there?''

Bib thought for a moment and then said, ``The first card can be any of 13 cards, any of the remaining 12 cards can come second, and any of the remaining 11 cards for the last card. Since we are performing successive actions, we multiply and obtain, well, you work it out 'cause you've got the paper.''

``Multiplying gives ways. But now I must issue the first, and most important, warning about counting. Always ask yourself if you have counted objects too many times!''

``I see what you're gettin' at, professor. I can get any particular hand dealt in six different orders. So I've got to divide that number by six, right?''

``That's correct, Bib. And you told me you were not much of a numbers man. You have some hidden mathematical talent. Upon dividing we obtain 286 heart flushes. But a three-card flush could be hearts, spades, diamonds or clubs. So we have to add 286 together four times and obtain 1,144 three-card flushes. Again we must remove the 48 straight flushes giving us 1,096 three-card flushes.''

Bib grunted and finally said, ``1,096 three-card flushes compared to only 720 three-card straights certainly explains why I noticed many more flushes than straights in three-card monte.''

Next time we shall consider all three-card poker hands.