Here’s kind of a tricky question if you’ve never thought about it: how many pips away from an attacking checker is the most dangerous, that is, the one that leaves your opponent the most rolls that can hit you (**shots**)? We know from other board games that 7 is the most likely number when rolling a pair of dice, so is the answer 7 pips away?

As a concrete example for when this information could be important, consider the position below:

Your opponent is camped out on your 6-pt and would *love* to hit your checker and win the game. You rolled a roll that allows you to place your checker with precision. You can leave it where it is 10 pips away, or you can advance it all the way to the 12-pt, where it would be 6 pips away. Where is it safest to leave it for 1 roll before you try and bring it home?

It turns out the most *dangerous* spot is 6 pips away!

..What!? How can that be?

Backgammon checker movement is unique among games in that you play the dice independently. That means if, next roll, either of your opponent’s dice has a 6, or if they *combine* to make a 6, your checker can be hit. Add in doubles and you have to throw out what you thought you knew about dice probabilities.

It turns out that of the 36 possible dice combinations, 17 of them can hit a 6. That is, you are leaving 17** shots**. 17/36 is nearly half the rolls! They are comprised of:

- Any roll where at least one die has a 6: 61, 16, 62, 26, 63, 36, 64, 46, 65, 56, 66
- Any roll where both dice add up to 6: 15, 51, 24, 42, 33
- Any doubles that happen to be factors of 6: 22

By contrast, a distance of 7 pips can only be reached by rolls from the second category, where both dice add up to 7, adding up to 6 shots: 61, 16, 52, 25, 43, 35. So it turns out to be 3 times more likely to be hit 6 pips away than 7!

The keen observer will recognize that there’s nothing special about the numbers 6 or 7; what’s notable here is that the numbers 6 and below appear on each die, and numbers 7 and above need some kind of combination of the two dice. These two sets of numbers even have different names — direct shots and indirect shots, respectively.

It’s actually worth it to eventually memorize the number of shots for every space away^{1}. At this stage, the important things to learn are that:

**Direct shots are***way***more dangerous than indirect shots**.**For direct shots, the closer you are the safer you are**(11 shots 1 pip away vs 17 shots 6 pips away)**For indirect shots, the farther you are the safer you are**(6 shots 7 pips away vs 3 shots 12 away)

The plot below further illustrates these points:

Leaving shots is unavoidable in a real game of backgammon. If you are forced to leave one, don’t leave a direct one. As we continue to improve our game we will learn to identify moments when it is worth leaving direct shots for your opponent. However, it’s good to know the risks involved.

##### Further reading:

- A video by GM Marc Olsen on dice combinations and shot-counting
- An article by Paul Stephens with more detailed examples of dice probabilities in backgammon

- Starting from 1 pip away, up to 12 pips away, the number of shots, in order, are 11, 12, 14, 15, 15, 17, 6, 6, 5, 3, 2, 3). ↩︎

Next lesson: The danger of higher points

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