Backgammon Strategy: Opening Moves, Blots, Primes and the Race

The fifteen opening rolls

The first roll of a game is never a double here — equal dice are thrown again — so the first move is always one of fifteen rolls. Opening theory was settled by computer rollouts of millions of games, and for most rolls it is not in doubt. Points are numbered as on the board: you move from 24 down to 1, the 13-point is your midpoint, the 7-point is the bar point and the 5-point is the golden point.

As it plays here, the computer follows theory on the rolls that make a point (3-1, 4-2, 5-3, 6-1) and on 6-5. Where it parts company, it never splits its back checkers. With 4-1 and 3-2 it plays 13/8, with 5-2 and 4-3 13/6: one checker onto a point it already owns, no blot at all. With 6-3, 6-2 and 2-1 it brings a single checker down from the midpoint (13/4, 13/5, 13/10) rather than splitting or bringing down two builders. Its evaluation charges every blot by its chance of being hit, and with one move of look-ahead that charge outweighs everything else. Theory goes the other way: early in the game a hit costs little, and a builder that may make a new point next turn is worth the risk. Given time to look two moves ahead, Expert comes round to the same view — with no time limit it plays the standard 13/8 13/11, 13/9 13/10, 13/10 13/11 and 24/23 13/8 — but on an ordinary computer its 0.8-second budget runs out before that search gets anywhere, and it plays Hard's move.

Easy chooses at random among plays its simple count rates as equal: with 6-5 it ran 24/13 in 24 of 60 trials, with 6-1 it made the bar point in 37 of 60. The last column is the page's own Expert code with the clock switched off, so that all eight of its candidates get the two-move search; that takes several seconds a move.

Blots: how many of the 36 rolls hit

A blot is hit when an enemy checker is exactly the right number of pips away and the roll contains that number — either on one die, or as the sum of both dice, or as a double played three or four times. Within six pips a single die is enough, and 11 of the 36 rolls contain any given number. From seven pips on, only combinations work and the count collapses. Counted exactly over all 36 rolls, with nothing in the way:

A combination shot needs an open point to stop on along the way. Your own made points between the shooter and the blot therefore take shots away. A few exact examples:

• Seven is the magic distance. Moving a blot from six pips away to seven cuts the shots from 17 to 6, almost three times fewer.
• Close is not safer. A blot one pip in front of an enemy checker is hit by 11 rolls; one six pips away, by 17. Inside direct range the risk grows with the distance.
• Two numbers are worse than one. Two blots at different direct distances are hit by at least 20 rolls of 36 — the 20 rolls that contain either number — before any combination is counted.
• A hit far from home costs little. A checker hit on your 22-point loses only three pips; one hit on your 4-point loses 21 and has to travel the whole board again.

Hard and Expert price every blot of theirs and yours with exactly the open-board column of the first table; they do not subtract shots for points in the way.

Points, primes and the bar

The points worth most are the ones that block: your 5-point (the golden point), then the 4-point and the bar point. They sit right in front of your opponent's back checkers, and each of them takes away a number those checkers need. That is why the best opening rolls are 3-1, which makes the 5-point, 4-2, the 4-point, and 6-1, the bar point. The more points you hold in your home board, the harder it is for a hit checker to come back in. This is exact:

How much these structures are worth shows in simulated games. Below: how often an event happened, and how often the side that got there first went on to win — in 60 games of Expert against Hard and in 4,000 games of Easy against Easy. A prime counts only with an enemy checker still behind it.

An anchor — two of your checkers on one point in the opponent's home — cannot be hit, and it makes a full closeout impossible: even if every other point of his home board is closed, you still enter with 11 rolls of 36, the ones that contain your anchor's number. An advanced anchor, on his 5- or 4-point (your 20- or 21-point), also takes away his best building points and gives your back checkers an easy way out; one on his 1-point still stops a closeout but blocks little else.

The race and the pip count

The pip count is the total distance your checkers still have to travel: for each point, the number of checkers times the point number, plus 25 for every checker on the bar. Both sides start at 167. An average roll moves 8⅓ pips (36 rolls, with doubles counted four times), so being the one to roll next is worth about half a roll — roughly four pips. Once the armies have passed each other, nothing but the race is left. We played 38,342 random no-contact positions to the end, both sides using the bear-off rule of the Expert level, and measured how often the side on roll wins:

So with equal counts the side on roll wins about 60 races in a hundred — the roll is worth 3–4 pips in races of 70–130 — and the lead at which it becomes a three-to-one favourite is about 10% of its count. The “8–10%” figure you may know from backgammon books is a doubling-cube rule: it marks where a race becomes strong enough to double, not where it is even. There is no cube here, so what you need is the table: ahead by a tenth on roll, you win about three games in four; behind by a tenth, about 35–45 in a hundred.

• Ahead in the race: break contact, run, avoid hitting unless it is free. Every exchange of hits gives the trailer another chance.
• Behind: keep an anchor or a back checker in his home and wait for a shot. Without contact, 20% behind with your opponent to roll leaves you about one race in eight.
• Counting quickly: count the difference rather than both totals — checkers on matching points cancel out.

Bearing off: wastage beats pips

Pips are not the whole story at the end. A checker on the 1-point needs a whole die to come off, whatever the die shows; three checkers on one point need three dice. We computed the exact average number of rolls needed to bear off, with perfect play, for home-board positions of up to fifteen checkers, and compared positions with their pip counts:

* Rolls needed × 8⅓, the average roll: the pip count the position really plays like. The difference from the real count is the wastage.

• A smooth board beats a lower count. The position with 54 pips spread over all six points comes off half a roll faster than one with only 45 pips piled on the 1-, 2- and 6-points.
• Bring checkers in on the high points. Coming home, put them on the 6-, 5- and 4-points rather than playing deep to the 1 and 2: deep checkers waste big numbers later.
• Fill gaps. While checkers remain above a gap, that number cannot bear a checker off and has to be played as a move inside your board.
• The last checkers: one checker on the 6-point takes 1.25 rolls on average; two checkers anywhere take at least one roll, and two on the 6-point take 2.11.

The levels differ sharply here. In 400 random bear-off decisions where the choice mattered, Expert, which switches to a race count, picked a play that is not the best only 9% of the time. Easy and Hard picked a worse one in 71–80% of them, about a fifth of a roll each time: they hold on to points in the home board as if contact still mattered — with 5-2, one checker on the 1-point and two on each of the others, Hard plays 6/1 6/4 and bears nothing off. Over a whole bear-off it adds up: from the smooth 54-pip board in the table, the best play needs 7.88 rolls, Expert 7.92, Easy 8.89 and Hard 8.90 — a full roll, about eight pips, given away.

How to beat each level here

All three levels see every legal way to play a roll; they differ in how they judge the result. Hard and Expert are closer than their names suggest. Expert's two-move search has a budget of 0.8 seconds, and the one-move stage that comes first takes most of it: on the computer we measured, in two contact decisions out of three Expert had time to deepen only its first candidate, which means it played exactly Hard's move. Where it really differs is the race and the bear-off. In 60 games against each other Expert won 31. The table shows how often a player with each style beat each level (who starts is decided by the opening roll, as in the game):

Against Easy, 2,000 games per line (4,000 for Easy itself). Hard and Expert take about a second a move, so against them only 6–60 games per cell were played: read those as rough counts, not percentages. Expert was run with a fixed clock that reproduces its 0.8-second budget on the computer we measured. Gammons and backgammons count as one win, as in the site score; random play lost 1,915 of its 2,000 games against Easy by a gammon or backgammon.

• Easy looks only at the race count and at blots, and hits whenever hitting gains pips. It does not value points, primes or a home board, so it is beaten by building: make your 5- and 4-points, prime its back checkers and pick them off. Hitting back alone, as the “always hit” line shows, only makes it an even game.
• Hard values points, primes and anchors and checks your every reply. It is weak in the opening, where it plays the quiet moves from the first table, and it prices blots without counting your blocking points. Take the tempo it gives you early: split, slot, build. And do not fear a race: it bears off about a roll worse than it could.
• Expert plays like Hard while contact lasts, and switches to a pure pip-and-wastage count once the armies have passed. It is hardest to outrun; your chances are in the middle game — a prime, an anchor, a well-timed hit — before the race starts.

The mistakes that cost the most

• Leaving a blot within six pips of an enemy checker when seven or more was possible: 11–17 shots instead of 6 or fewer.
• Playing a pure race when behind. Without contact, a tenth behind with the opponent to roll wins only about one race in four.
• Hitting for the sake of it when ahead in the race: every exchange of hits gives the trailer another chance.
• Stacking checkers on the low points while bearing in: a position can be ahead on pips and still lose the bear-off.
• Giving up the anchor too early. While you hold a point in his home, you cannot be closed out.