| One of the interesting things about doing gaming analysis is that every now and then I discover a clever piece of mathematics that is just the thing I need to complete a job. Unfortunately, I don't always discover the mathematics I need. A year or so ago I was asked to look at a video game that was based on both geometry and Poker hands. Try as I did, I just couldn't see a way to analyze the game nor did I recall any mathematics that I felt would help me out. To this day I'm baffled by this game and occasionally I go back and take another look at it -- frustrating. Happily, this doesn't happen that often, usually things work out. Here is just such a story. I am writing a book about state lotteries which should be published sometime this spring. The title is The Lottery Book, The Truth Behind the Numbers and it is being published by Bonus Books of Chicago. The Foreword to the book is by Frank Scoblete. In it I analyze just about every state lottery game in the country and explain to the reader just how such analyses are carried out. As you might imagine, I ran into some unusual games whose analyses required some serious thought. To my mind, the most notable of these was provided by the state of Wisconsin. Wisconsin has a lottery game called City Picks. Here is how it is played. The player is provided with nine Wisconsin cities: Chippewa Falls, Dodgeville, Green Bay, Kenosha, Madison, Milwaukee, Superior, Two Rivers, and Wisconsin Rapids. To play City Picks the bettor simply arranges these nine cities in order 1 through 9. The Wisconsin State Lottery then randomly arranges these same cities in a different order, again 1 through 9. The player is paid according to how many of his numbered choices match the lottery's numbered choices. Since I cover the game in detail in my book, I'm not going to repeat the complete analysis here. Suffice it to say, however, that in order to carry out such an analysis one must determine how many of the 9! or 362,880 permutations of these cities occur as no matches, one match, two matches, and so on. Let me show you an example. Suppose for a given lottery ordering we want to determine how many ways we can produce an ordering of ours that has exactly 5 matches. Well, we first have to choose which 5 of the 9 numbers will be our matches. This is old stuff to us by now; it is just the number of way of picking 5 things from a set of 9 and is the number 9!/(4!5!) or 126. Now for each such choice the remaining 4 numbers have to not match those of the same remaining 4 lottery choices. In mathematical jargon, they have to be a derangement of the lottery's choices. Let's take a closer look at this. Consider the integers 1 2 3 4 in that order. How many ways can we rearrange these numbers so that there are no matches in the rearrangement? For a small number like 4 this is relatively easy, we can just list them. The derangements are: 2 3 4 1 3 4 2 1 2 1 4 3 2 4 1 3 4 1 2 3 3 1 4 2This past September I had the pleasure of attending the Global Gaming Expo, or G2E as it is known, that was held in Las Vegas. The convention was fascinating, as usual, with all of the new games, gaming machines, and gambling-related products on display. This year, for me, there was some added excitement. First the gaming writer Jean Scott and her husband had arranged a reception for gaming writers at the new Palms Hotel and Casino. It was held in early evening at the Ghost Bar high atop the Palms. There was an open bar, delicious hors d'oeuvres provided by owner George Maloof, and a spectacular view of Las Vegas from an open balcony. It was really fun to to finally meet other writers who, up until then, had just been names on a page. Jean, if you're reading this, thank you for all of your efforts at putting that fabulous mixer together. Thanks as well to George Maloof for his part in the festivities. Second, my friend and colleague Frank Scoblete was attending G2E and we happened to be staying at the same hotel. Although I had met Frank in person on an earlier occasion, most of our conversations are by email. This time we were able to spend some time together in person and also do some gambling together. It was this latter activity that led to my third dose of added excitement. Frank, John Grochowski, and I had dinner together and afterwards Frank and I headed for the Craps table. Frank and I were merrily shooting Craps when all of a sudden Frank picked up his chips and said to me, "Move to the next table; the best dice shooter in the country is there" and away he went. Well, I had a bet on the Come so I had to wait for it to be resolved before moving. It seemed to take forever. Finally I moved in next to Frank at the adjoining table and saw the shooter to whom he was referring. The guy was located to the left of the stick man and his aura was pure concentration. He would carefully set the dice, deliver them smoothly, and they would come to rest after gently hitting the end of the table. His concentration was intense. Lots of points were rolled by this gent. When he finally did seven out, the players at the table simply passed the dice right back around to him. It was an experience. I noticed the box men becoming antsy and I guess the shooter did too because he finished his roll and stepped away from the table. Frank turned to me and said, "You have just witnessed Sharpshooter. Come on, I'll introduce you." Now I'm sure there are skeptics among you who wonder if there really is a guy named Sharpshooter and if he really can control the dice. I can now tell you from personal experience that he certainly does exist and because I saw him in action I'll say it clearly: He is one terrific shooter. Not only did I meet Sharpshooter but I met his colleague Long Arm, another controlled shooter. Both of these guys were very personable and we had a really pleasant conversation; I sincerely hope our paths cross again. One of the things Sharpshooter told me is that he had written a book about Craps that was about to be published and that he would see to it that I got a copy. True to his word a copy of Getting the Edge at Craps, How to Control the Dice was in my mailbox a few weeks later. |
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