Backgammon

You move your fifteen checkers around the twenty-four points towards your own home board, and off the far edge. Throw two dice: the two numbers are two separate moves, and a double gives you four moves of that number. A checker may land on an empty point, on one of your own, or on a point holding exactly one enemy checker — which knocks that checker onto the bar. Once every one of your checkers is home you start bearing them off, and the first player to remove all fifteen wins the game.

Both sides begin on 167 pips — the total distance every checker still has to travel. Watching that number, rather than the checkers, is how experienced players know whether they are racing or fighting.

The cube is what turns backgammon from a dice game into a gambling-theory problem, and it is the part almost every free implementation leaves out. Before you roll, if the cube is in the middle or on your side, you may offer to double the stakes. Your opponent then either drops — conceding immediately for the stakes as they stood before the double — or takes, doubling the value of the game and taking the cube, after which only they may double next.

That last clause is the whole game. Handing the cube over gives your opponent a weapon, so a double is never free. In a money game the break-even point for taking is a 25% chance of winning; owning the cube afterwards is worth a few points of that, which is why strong players take a little below 25% and why the computer here takes down to 22%. It offers the cube when it makes itself between 68% and 88% to win — below that a double is premature, and above it the game is worth more played on for a gammon than cashed in.

This is the rule clones get wrong, and it costs games. Once all fifteen checkers are in your home board, a number takes a checker off the point it matches: a 4 removes a checker from the 4-point. A number larger than the point it is played from only bears a checker off when there is no checker left on any higher point. If you throw a 5 with checkers on the 6-point and the 2-point, the 5 must be played inside the home board as 6/1 — it cannot lift the checker from the 2-point, because the 6-point is higher.

The consequence is practical: leaving a straggler on your 6-point while the rest of your checkers crowd the 1-point and 2-point wastes huge numbers of pips. Spread the checkers evenly across the six points as you bring them home and every number you throw does useful work.

A lone checker on a point is a blot, and how exposed it is depends almost entirely on the distance to the nearest enemy checker. Anything within six pips can be hit by a single number, and each number appears in 11 of the 36 possible throws. Combinations fill in the rest. The board below is the count this engine computes, and it is the same table every backgammon book prints.

Six pips away is the worst place to stand: 17 of the 36 rolls hit it, because a direct 6 is joined by 5-1, 1-5, 4-2, 2-4, 3-3 and 2-2. Move one pip further and the danger collapses — at seven pips away there is no direct shot at all and the count drops to 6 of 36. Every point you own in between kills the combination shots that pass through it, which is why a blot behind a wall of your own points is much safer than the raw distance suggests.

A checker on the bar must re-enter in your opponent's home board before you may move anything else, and it enters on the point matching the number thrown: a 3 enters on their 3-point. If that point holds two or more enemy checkers, the number is dead. With five of their six home points made and one open, only 11 of the 36 rolls get you in; with all six made you cannot enter at all and the entire roll is forfeited, turn after turn, while your opponent plays on unopposed.

The arithmetic is 36 minus the square of the number of shut points, which is why the fifth point made is so much more valuable than the second. Closing your home board while an enemy checker sits on the bar is the most crushing position in the game, and it is worth going a long way out of your way to reach.