Your opponent rolls a 1–2 and makes another point. A few turns ago, a 1–2 gave you nothing you wanted. It is easy to remember the two rolls as good luck for them and bad luck for you.

Looking at the checkers before the roll helps explain why the same dice can be useful for one player and awkward for another.

Give both positions the same task

We made two positions for this example. Both come from legal ways to play the same 3–2, starting from the same board. Both sides have the same distance left to travel in either version.

To compare them, leave Red’s checkers where they are and give Ivory another roll. Set a small, clear goal: make the 4-point while keeping the 5- and 6-points. Making a point means having at least two of your checkers on it, so your opponent cannot land there.

In the first arrangement, Ivory used the 3–2 to bring a checker from 11 to 6. That leaves four checkers on the 6-point and two on the 5-point.

In the other, Ivory moved 8 to 5 and 11 to 9. Now the 5- and 6-points each have a spare checker available, with more checkers behind them.

Now roll 1–2

In the second position, play 5 to 4 with the one and 6 to 4 with the two. This makes the 4-point and leaves two checkers on each of the other points.

Try that in the first position. Moving 5 to 4 would leave only one checker on the 5-point. You can make the new point, but you cannot keep both old ones with this roll.

The dice were the same, but having spare checkers made the difference.

Give the same dice to both positions

The goal is to make the 4-point while keeping the 5- and 6-points. Choose a roll and play through an example. Both arrangements followed legal plays of the same 3–2.

5 of 36 rolls can make point 4 and keep points 5 and 6 (13.9%).

One way to play the 3–2.

12345678910111213141516171819202122232400

Ivory moves toward 1; Red toward 24. All point numbers are shown from Ivory’s side.

Read the position

Ivory, point: number of checkers. 5: 2, 6: 4, 8: 3, 13: 4, 24: 2. Red: 1: 2, 12: 4, 17: 3, 19: 4, 20: 2. On the bar: Ivory 0, Red 0. Borne off: Ivory 0, Red 0.

Count all the ways it can work

A single convenient roll could make the comparison look better than it is, so the example checks all 36 possibilities.

In the stacked arrangement, 5 rolls out of 36 can achieve the task. In the other, 15 out of 36 can do it. That is about 13.9% compared with 41.7%.

These are the chances of completing our particular task, not the chances of winning the game. Other rolls may offer a hit, an escape or a useful move elsewhere.

The second arrangement offers more possibilities overall, but it does not improve every roll; try the unhighlighted squares too.

Prepare more than one useful roll

A checker placed behind the points you want to make is often called a builder. Keeping builders at different distances can let several combinations work. A spare checker also lets you move without immediately giving up the point it came from.

Before confirming a play, imagine a few ordinary rolls on your next turn. Could several of them make a point, cover a blot or help a back checker escape, or would you need one particular combination?

A strong position can make an ordinary roll look fortunate. You still need the dice, but you can prepare more places for them to help.

Flexibility, builders and diversification are established backgammon ideas. Kit Woolsey describes the aim of creating more useful rolls in the opening of his article on duplication, which informed this piece. We created the paired positions, the particular task and the roll-by-roll experiment for CUBE Backgammon.

What the percentages measure

For every roll, we check all the ways to play the full turn legally. A roll succeeds if at least one finishes with two or more ivory checkers on each of points 4, 5 and 6. Doubles give four moves, and the normal forced-dice rules apply. The previous 3–2 uses five pips in both versions; the opponent’s position is unchanged. We do not use this narrow task to claim that one complete position is objectively better or to estimate a player’s winning chances.