Aces and Knaves

tonight."
    He put his hand on the shoulder of one distinguished-looking gentleman who was playing craps with a beautiful but inadequately-covered woman beside him and said, "Jed, when Sally rolls the dice don't let her bend over too far or we'll have to put her assets back into her dress. I'd better tell one of my assistants to get a warm spoon ready."
    When I had a chance I asked, "Why don't you have slot machines?"
    James led me to one side of the room and said, "First, there is no skill in playing the slots. They're all luck. I only like games and puzzles with at least an element of skill. All the games played here fit into that category. Second, as you may have noticed, we don't use money here."
    I didn't want to sound as if I were from Buttonwillow, but I didn't know how else to phrase the question. "Are you telling me all those chips don't represent money?"
    James smiled an engaging smile and said, "When you've acquired a certain amount of wealth you can do pretty much what you like. What I like is games and puzzles. Why shouldn't I be able to set my basement up as a casino and invite my friends over, if I want to? What game would you like to try?"
    My skepticism at his answer boiled over, but I didn't know what else to say. For one thing, the players were concentrating awfully hard for nothing being at stake. In any case, why not try a game? With no chance of losing money I couldn't get into trouble. A little blackjack, perhaps? No, I really needed to ask James some questions about Ned. We were standing beside a table with a backgammon board on it. I said, "Do you play backgammon?"
    "I play a bit of everything. Would you like to have a go?"
    We sat down and arranged our fifteen checker-like pieces on the designated points. As we each rolled one die to determine who would start I asked in what I hoped was a casual manner, "Does Ned come here often?"
    "Whenever he's in San Francisco. Ned's an old friend of mine. We go way back."
    I rolled a six; James rolled a one. Using these rolls for my first play, I made my bar-point, or seven-point; that is, I moved two pieces to it, creating a block.
    "What games does he like to play?"
    James rolled a 3-1 and made his five point.
    "Oh, he likes to shoot craps or play blackjack. Sometimes he plays poker."
    I rolled a 4-3 and moved two pieces to my side of the board from his twelve-point.
    "Would you say he is a compulsive gambler?"
    James rolled a 6-3 and moved a piece from my one-point, hitting one of my piecess and sending it to the bar.
    He sat, looking at the board, as if studying the game. I commanded my hand that held the dice cup to be still as I waited for his answer. He finally looked at me and said, "A year ago I would have said there was nothing compulsive about Ned. Now I'm not so sure."
    "Any special reason?"
    "Because of things that have happened."
    An enigmatic response, but I had better not push it any more or I would arouse suspicions. I rolled a 3-2, usually not a great roll, but I got my piece off the bar with the two and used the three to hit James' piece.
    "If you don't play for money, what's the thrill?"
    James smiled a quick smile. "The thrill of playing any game, I guess. Trying to beat your opponent. Or the dice. Or the cards. Trying to excel. And we do keep rankings in each game, from the biggest winners to the biggest losers over the course of a year."
    That was still unsatisfactory, but I didn't ask any more questions. As the game proceeded, James made what I considered to be several tactical blunders in how he moved his pieces. However, the game was still undecided down to the last two moves.
    James rolled a 5-1 and bore a piece off the board with the five. He now had only two pieces remaining, on his two-point and his three-point, and he could move one of them one point. To my surprise, he moved the piece on his three-point to his two-point, leaving two pieces there, instead of from his two-point to his one-point, which would have left them on his one and three.
    There are 36 possible rolls with two dice (six times six). With two pieces on his two-point, there were ten rolls that wouldn't move both his pieces off the board on his next turn; they included every roll with a one in it except a double one, since doubles count double. But, if he had left the pieces on his one and three-points, there were only two rolls that wouldn't have cleared the board for him: 2-1 and 1-2.
    I won the game because of his mistake.
    James congratulated me and said,