...Secrets and Subtleties

Among the games people play, backgammon is a unique combination of the very old and the very contemporary. While the game itself dates back thousands of years (backgammon boards were found in King Tut's tomb), the modern gambling version was developed only a generation ago. The manner in which backgammon is currently played evolved only in the past ten years, and the evolution is continuing. Prior to the introduction in the Twenties of a doubling feature, the game was hardly more than an obscure relative of Parcheesi, played in British clubs and by natives of the Baltic and Mediterranean regions, where it has been popular for millennia. The doubling feature (described below) made it a gambling game par excellence. As time passed, the game naturally attracted gamblers par excellence. These gentlemen (and ladies), most of whom are still alive, playing and prosperous, re-examined the game and initiated an entirely new style of play, based on game and probability theories, which, while their basics have been known since Pascal, were seriously investigated only after World War Two. The connection has never been established, but the parallels between the evolution of contemporary backgammon and the development of the high-speed computer are too striking to ignore.

Backgammon is one of the most difficult board games in existence and the most difficult game of chance. While its rudiments can be learned in a few minutes, pursuing its subtleties can consume a lifetime. The Romans called it ludus duodecim scriptorum—the 12-line game. They played it, as it can be played today, with an equal number of shells and pebbles and 12 lines scratched in the dirt. In the modern game, the lines have given way to triangles, which many people think are the conventional decoration for the backs of checkerboards. But backgammon, essentially, is still a game of 12 lines.

It's normally played by two people (though more can play, in what is called a chouette) on a board marked with 24 narrow triangles arrayed in two rows of 12. The triangles, which backgammon players call points or pips, are usually in two alternating colors, but the colors have no significance, other than to make it easier to count the pips. Each player has 15 checkerlike disks (called counters or men or stones) that are placed on the board in a predescribed pattern and moved from point to point as determined by the roll of a pair of dice. The roll of a 6–5, as an example, entitles the player to move one of his men six pips and another, five; or he can move one man all 11 points. Doubles count twice what the dice show. Thus, a 2–2 permits four moves of 2 points each.

Simply stated, the object of the game is to move your men around the board, into your home table (see photo) and then off, before your opponent can do the same with his. If you succeed in getting all your men off before your opponent has begun removing his, you achieve what is known as a gammon, which wins you double the stakes. And if you can bear off all your men while your opponent still has a man stranded in your home table, you win a backgammon, triple the stakes. Since the players must move their men in opposite directions, the development of the game resembles the clash of two armies in battle. Both players begin the game with soldiers arrayed on the four quadrants of the battlefield. Each player's troops must pass through the enemy's lines—and risk being captured on the way—before they can reach home safely. And even home is not necessarily safe, because the enemy might have left a few soldiers waiting to ambush whatever stragglers have failed to attach themselves to the main body of their army.

The photograph at left shows how the counters are arranged at the start of play. In a sense, a backgammon game is one third over before it begins, since each player opens with five counters in his home table, on the 6 point. The game can be played with most any beginning arrangement, and many variants exist, but the starting arrangement shown here (or its mirror image) produces the best and most interesting games and is now used almost universally. After moving all 15 of your men into your home table (points 1 through 6), you can begin removing them from play, also according to the throw of the dice. The throw of a 5–4, for instance, would permit the removal of one counter from your 5 point and another from your 4 point. The winner is the first player to bear off all his men. Since the players roll the dice alternately, ties cannot occur.

At the heart of the game is the element of capture. Each player can move any of his men to any point that his dice and the direction in which he is moving permit, unless the point is already occupied by two or more of his opponent's men. Having two or more men on the same triangle "makes the point," securing all the men on that point from capture. Each player may jump over a secured point, but he can never land on it. A lone man (called a blot) is captured if an opponent lands on the point on which it rests. Captured men must start over again, re-entering the board in the opponent's home table, also according to the dice, in a process that is rather the opposite of the bearing-off procedure that concludes the game.

The subtleties of backgammon become apparent when you realize that it's not only impossible but unwise to keep your men safe at all times. Open men create wider options and a more fluid attack. Much of the skill of backgammon lies in understanding when and where to place your open men so that your opponent has the least possibility of hitting them. Each configuration on the board requires a new set of computations. Should you play a blocking game to keep your opponent from moving home easily? Or would it be better to take the offensive and run?

In many situations, the most critical decision a player can make is whether or not to capture an opponent's man. Capturing takes the man off the board, slowing the opponent's progress, since the man must re-enter at the beginning, in his opponent's home table, and all captured men must be re-entered before any of their teammates can move. Getting hit early in the game is of little consequence and may actually help you. But as the game matures, re-entering a captured man becomes progressively more difficult, because the opponent, bringing his own men home, blocks more and more of the re-entry points. On occasion, a captured man cannot re-enter at all, because the opponent's men block all six points (called a closed board). When that happens, the blocked player simply seethes politely, abandoning his turn at the dice until an opening appears, whereupon he can roll again.

If someone were to ask The Rand Corporation to design a perfect gambling game, the result would probably be backgammon. Because captured men re-enter at the very place where the victorious player is bearing off his men, the possibility exists that a loser can pick off one of the victor's remaining men and turn the whole game around in one roll of the dice. Backgammon is quick (a typical game might last seven minutes), so there's always another game in which to recoup. The doubling feature increases the stakes in a way that is psychologically acceptable to both players and encourages the abandonment of dull or hopeless games. The one-on-one nature of the game (as opposed to casino gambling, where players all compete against a monolithic house, which has the odds in its favor) fosters camaraderie. And the ever-present rapid-fire roll of the dice provides the tempting (but erroneous) suggestion that backgammon is the ultimate game of chance. This is worth some discussion. The pastimes that we call games fall into two categories: games of chance and games of skill. Both categories embrace a wide spectrum of superficially dissimilar pursuits. Ice hockey and chess, for instance, are very different games, but both are games of skill. Playing bridge has little in common with buying a lottery ticket, but both of these are games of chance. The sole distinction between the two categories is that games of chance employ a randomizing agent. It might be a wheel that spins, cards that are shuffled and dealt, dice that are thrown onto a table or numbers that are plucked from a hat. Whatever it is, the players have no control over it. That part of the game is in the hands of the Fates.

Or is it? During extended play, good bridge players will almost invariably triumph over inferior players, no matter what cards they hold. They win because they know more about the play of the game and because the randomizing element—the fall of the cards—is ultimately less important than how the cards are played once they are dealt. Skill plays a large role in bridge, and in all challenging card games, because the influence of the randomizer is relatively small.

The subtle glory of backgammon is that it lets the randomizer run rampant. A typical backgammon game involves perhaps 50 rolls of the dice. A typical match, three or four hours long, might entail 2500 rolls. And a typical big-time backgammon player, playing a match or two a day, will see 2,000,000 or 3,000,000 dice rolls in a single year. On the surface, this would make backgammon the ultimate game of luck. But anyone acquainted with the laws of probability knows that the facts argue otherwise. Over time, dice roll predictably. If you are familiar with probability, and if you are throwing the dice millions of times a year, they will roll very predictably.

This sort of chance that is no longer chance exists all around us. The air molecules in the room in which you sit are moving rapidly and at random. Theoretically, nothing prevents them all from darting out the door, in which case you would suffocate. But they don't, and you can reasonably expect that they never will. There are just too many of them. Random motion (with molecules (continued on page 166)Backgammon Secrets(continued from page 122) it's called Brownian movement) is subject to the laws of probability. And when you are dealing with large numbers of random movements, the laws of probability are as immutable as gravity itself. The distinction between 1023 air molecules and 3,000,000 rolls of dice is one of degree but not of kind. The same principles apply in both cases. The superior backgammon player might lose an individual game, or a whole string of them, to the rankest of amateurs. But as long as he respects the laws of probability, assuming he plays enough, he is bound to win, overall. He would be breaking the law if he didn't. In fact, the only time he would face trouble would be against opponents as knowledgeable as himself. This actually happens, in the major tournaments that are taking place with ever more frequency and ever larger participation, at jet-set hangouts the world over. And the results are just what you would expect: When 20 or 40 of the best backgammon players in the world get together, no one can say who will win. Even the best of the best has never won more than three major championships in a row. And such a winner, if he has a decent respect for reality (which he surely must, given his knowledge of backgammon), would have to acknowledge that he won because he was slightly luckier than his mates. But only at this level is luck involved. When the pro plays an amateur, even for a few hours, the chances of the pro's losing are comparable to the chances of his suffocating due to a sudden exit of air.

If you can recall your feeling when you first learned how to beat your little sister every time at ticktacktoe, then you are privy to the secret smugness of the big-league backgammoner in a money game with an amateur. For every roll of the dice, the pro knows the best response. That doesn't mean that he won't adjust his play to suit his opponent; he will. And experts frequently disagree on the best move. But in a game between an expert and an amateur, there is little need to worry about such refinements. The pro will roll his dice and move his men immediately. He doesn't have to examine the board, because he already has taken into account how he will play any of the 21 different combinations he could possibly roll. (Two dice with six faces each produce 36 combinations, but only 21 of them are distinctive; the rest are duplicates, such as 6–1 and 1–6. Thus, the pro must account for the six doubles he might roll, plus the 15 combinations that are not doubles, for a total of 21 possible rolls.) He knows that he will roll 5–3 twice as often as 5–5 (because there's only one 5–5, while there are both 3–5 and 5–3) and plans his game accordingly. He knows that he will roll a 6, in one form or another, more than he will roll any other number (16 of the 36 combinations show a 6 or add up to it, and 2–2 also permits a move of six spaces, since doubles are moved twice the amount on the dice) and this, too, governs the deployment of his men. In every move he makes, he places his men so as to take advantage of what his next roll and his opponent's are most likely to be. Every time he moves his men, he is maximizing his chances of winning. The more he moves, the more likely he is to win.

Of course, if you're playing against him, this won't be immediately obvious. If you're playing for high stakes, he probably won't want to show you how good he is. And even if you're just playing for the hell of it, some of his moves will seem to defy common sense, and all of his moves will be made very quickly. He doesn't have to count the triangles, because he's made such a move before, thousands of times. He moves his counters with the hypnotizing fluidity of a shell-game operator. He is intimately familiar with these warm and silk-smooth disks. They clunk together in sounds and patterns he has known all his life. His dice knock noisily in his cup and strike the board, hard, on his right. Even before the cubes settle, his hands have assumed a life of their own. He is a master weaver, he is touch typing—he is moving his men quicker than you can count. And all the while he is talking, quietly, over his shoulder, to an insouciant braless blonde about a chouette they played in Biarritz a fortnight ago. All in all, it's a dazzling performance. Lucrative, too.

To understand backgammon the way this man does, you must clearly appreciate its paradoxical nature. Backgammon is a game of chance, but one in which chance plays such a large role that it approaches predictability. The expert minimizes chance, because he knows that chance is a matter of probabilities and that probabilities have a habit (if you play enough backgammon) of turning into certainties—even into annuities.

The only comparable gambling game is craps, but craps is not a zero-sum game. The house has an edge, takes a cut. Theoretically, the more you shoot craps, the more you are bound to lose. True, the house edge is relatively small, so that superior crapshooters have been known to take money away, at least temporarily. But, as the saying goes, it's still a crap shoot. Backgammon belongs in a higher league. It's a zero-sum game: no house cut. Everything that one player loses, the other must win. If you are a craps player who occasionally fantasizes about that big table in the sky where the odds are precisely even, then you should be playing backgammon.

The probability paradox is precisely what makes backgammon such an attractive (and deceptive) gambling game. If you spend a few hours across the board from a superior player, you will be amazed, even outraged, at his egregious good fortune. He seems always to make the right rolls: double sixes, just when he needs them to march his back men halfway around the board; double ones to fill the two key points on his side of the table. Even junk rolls, such as 5–2 or 6–3, manage to work in his favor, instead of against him. It's enough to make you—well, to make you want to keep playing him. Surely his outrageous good luck can't last forever. Any time now, it's bound to turn your way, and you'll win all your money back and then some. But that road leads to madness, if not bankruptcy. You'd do as well betting that the air will leave the room. Yet there are people in this world—not many, but enough—who lose $100,000 a year at backgammon, year after year, and keep coming back for more. They would better spend their cash on a probability tutorial.

Backgammon is really two games: the play of the men and the use of the doubling block. As noted, the introduction of this device transformed the game. The doubling block is a cube of an inch or so. On the six faces are inscribed the progression of 2 doubled: 2, 4, 8, 16, 32 and 64. The block can be brought into play at any time. It is usually employed whenever a player thinks he has the advantage. For example, in a game being played for a dollar, when one player believes he is ahead, he might double his opponent to two, placing the block with the 2 on top. If the opponent thinks he is likely to lose, he can retire, thereby ending the game and forfeiting a dollar. If the thinks he is ahead or only slightly behind, he might accept the double. The stakes would then be two dollars and the other player would have no more right to double. However, if the man who accepted the double finds himself ahead later on, he can redouble to four. His opponent would then have the same options: He could abandon the game (and two dollars) or play on for four dollars. On and on it goes, with the right to double shifting from one player to another, until a double is refused or the game is won. In big-time play, many games are never played out. And in very close games, the doubling cube can move almost as fast as the dice. What began as a $100 game has been known to double and redouble to $25,600: here a $100 set-to can conclude with over $75,000 on the line, since the ultimate stakes are tripled if an opponent is backgammoned. Obviously, knowing when to accept a double and when to retire is essential to playing the game. There is a pokerlike element here, since if you tend to decline doubles, your opponent will pick up on this and offer you doubles from weaker and weaker positions, until you find yourself giving him money in games that you should have won. Conversely, if you tend to accept doubles too readily, your opponent will begin doubling from greater and greater strength, until you find yourself losing your money twice (or four times or eight times) as quickly. The pros can use the doubling block so well that even if they abandon three games out of four, they make it all back—and then some—in the few games they win.

The best book on contemporary backgammon (The Backgammon Book, by Oswald Jacoby and John R. Crawford) is quite reasonably dedicated "to the genius who invented the doubling cube and made backgammon the game it is." His name is not known, but his legacy endures. Backgammon's popularity today is growing as never before. The only comparison is with the Twenties, when contract bridge emerged from whist and took America by storm. Whether backgammon will go on to share bridge's staggering popularity remains to be seen, but early indications are favorable. Sales of backgammon gear—boards, dice, cubes, counters, etc.—have doubled each year for the past two. In November 1972, the game was given a professional status it never enjoyed before, with the formation of the Backgammon Association of America, a nonprofit organization, with offices in New York and Los Angeles, dedicated to improving the game and spreading its popularity. The group is headed by Prince Alexis Obolensky, a well-known figure in backgammon circles and prime mover behind most of the big-time tournaments held in the past decade. A few years ago, such events took place only once or twice a year, but already the B.A.A. is talking about a major tournament every week. Besides organizing such get-togethers, the B.A.A. will codify rules, sanction regional play-offs, set up an annual world championship (possibly televised) with substantial cash prizes, sell backgammon gear, award master points (as is now done in bridge) and even publish a newsletter. Playboy Editor-Publisher Hugh M. Hefner, a backgammon freak and no mean player, is a B.A.A. vice-president. On other fronts, private clubs dedicated to backgammon play have begun to spring up in places other than London and New York. Pips, a private social club in Beverly Hills, recently opened its doors to a wellheeled group of players and celebrities (who can relax in a discothèque when they're not calculating probabilities) and similar clubs are on the drawing boards for Chicago, Dallas and other cities. All in all, backgammon seems a game whose time has come; not bad for something 5000 years old.

Backgammon for beginners: The colors of the triangles (called points) have no significance. An empty point, regardless of color, goes to the player whose dice let him put two men on it. This board is set up for the start of play; points are numbered for convenience. You play white and sit at bottom. Your mission is to move your men counterclockwise around the board (arrows), into your home table and then out of play, before the enemy, moving clockwise, does the same. Roll your dice.

Your 11, 10 and 9 are points to build on. To avoid capture, you need two men on a point, but this is not always possible or even desirable. Strategy is everything.

Your bar point is crucial, because it's six points past opponent's 12, from which you must eventually move five men. Making the bar point early (with an opening roll of 6–1, for instance) creates a landing spot for your back men and blocks three consecutive points, hindering opponent's back men.

Enemy's 7 point is called his bar point; it's next to the bar that divides the board. Rolling 6–6 early makes this point: Move two men from enemy 1.

Opponent's 5 point is a strong defensive position if you can occupy it with your back men. An early roll of 4–4 can secure this key point for you.

Your two back men start out here, on opponent's 1 point; getting these critical men home is much of the game. See accompanying text for more details.

Your 5 point is critical. You begin with five men on your 6 point, so if you make your 5, you own two points in your home table. The best opening roll, a 3–1, secures your 5.

Your 4 point: also good to occupy in the early stages of the game, though better players will strive to make their 5 first.

3,2 and 1; avoid making these until you have made your 5 and 4. Making lowest points too early limits your manpower thus restricting maneuverability.