Some Backgammon Number Crunching

Doing the Maths: Your Best Strategy for Backgammon

  • The London Open… lots of bean counters
  • Pips... 167 and counting
  • Why thinking about your moves matters
  • Hitting a number with one dice
  • Hitting a number with two dice
  • Leave a long shot, not a short shot
  • A fluke

 Hollywood greats Cary Grant and Randolph Scott, 1932.

Hollywood greats Cary Grant and Randolph Scott, 1932.

Meeting the number crunchers… A lot of strategy for backgammon is about counting. If you get your head around the numbers, you will steadily improve. I wasn’t thinking about the numbers and didn’t know I needed to when I had some early success in reaching the final of a tournament at White’s in London. This was the first time I met and spoke with Phil Simborg, a world-class professional teacher and player from Chicago.

Phil is funny and kind, but he always ‘tells it like it is’ . We played a game or two and he said, ‘Simon, I wanted to play a game with you to determine whether you were lucky to get as far as you did at the White’s tournament or if you’re a really strong player.

Let me tell you, you’ve been lucky… you need to work at this!’ I gleaned many pearls of wisdom from Phil. He advised me to play more tournaments and online matches, to play against a good computer programme, and to read some books about the game. To get more live practice I entered the London Open. I was asked whether I wanted to join the professional or the amateur draw. Technically I was an amateur, but I couldn’t resist the opportunity to go up against professionals and the subsequent right to brag about it unreservedly if I did well, so I decided to enter both. I perversely relished the possibility of being knocked out twice in a day. 1

As I entered Brown’s Hotel, there was an apprehensive atmosphere and plenty of adrenaline-fuelled good humour. Backgammon players are a friendly crowd even when the stakes are high and this tournament was no exception. I was struck by how academic and studious some of the players seemed; we could have been in a university maths department common room. Indeed, it turned out that many of the players were highly skilled at maths. Chartered accountants and taxi drivers alike were rapidly crunching numbers, calculating the odds.

In the professional draw, a couple of players wore headphones throughout their match. I assumed they were listening to music, but a fellow competitor explained to me that they were merely blocking out the noise of their opponents. I was struck by how focused they were, they weren’t there to make friends or have a chat (which, for me, is often half the point of a good backgammon match), and they certainly weren’t there to have fun, although many other players clearly were.

One competitor in the amateur draw told me about her successful book, which outlined a new technique for coaching maths to children. I didn’t understand her explanation of it even after she repeated it, so I told her I hoped I wasn’t playing one of her young pupils in the first round. It wasn’t my day in the end; I lost early on in the amateur contest. In the professional draw, I was pleased that I managed to beat an accountant in the first round, but then I was knocked out in the second round by an economic forecaster.

That day I learnt that, while I’ll never be a walking calculator like some of those brilliant players, learning to think about the numbers in the right way is a great strategy for backgammon that will help anyone to become a winner at backgammon.

Having a good idea of who is ahead in the pip count — which measures the number of moves each player is away from bearing all their counters off the board — and by how much, will help you make better decisions and better inform your strategy for backgammon.

Have a look at the board’s starting position below. At the start of the game, the pip count is 167 pips for both players.

 Figure 17: Each player is 167 pips away from finishing.

Figure 17: Each player is 167 pips away from finishing.

Here’s how the pips work at the start of the game. If your opponent offered no resistance and you were never hit or blocked, you would need 167 pips to move all the checkers around and off the board.

We can count these 167 moves as follows.

Your two backmarkers are on your 24-point, so that’s 48 pips needed to bear them off the board. Your five midmarkers are each 13 pips away from being borne off, so that’s 5×13, or another 65 pips. Next are the three checkers on your 8-point; they will take 8 pips each to bear off, making another 24 pips. And finally the 5 checkers on your 6-point in your home board need 6 moves each to be borne off, so that’s 5×6, in other words 30 pips. The grand total is 48+65+24+30, which comes to 167 pips.

When you play backgammon against a computer program or online, you can adjust your settings to show the pip count. This is extremely useful because it tells you, at any point in the game, who is ahead in the race to finish, and by how many pips. If you’re playing on a conventional board you should learn to keep a mental estimate of approximately how many pips you have left.

In a live match you won’t have time to count the exact pips but it’s important to develop a feeling for how much you are winning or losing by. There are many complicated ways to do this, but the simplest is called a cancellation count, in which you don’t bother to calculate the total pip count but merely estimate one side’s lead. To do this, quickly compare the positions of your checkers with the positions of your opponent’s. Cancel out any checkers in the same position. Just count the ones in different positions and compare them. That should show you if, for example, you are ahead by 9 pips or your opponent has a lead of 12 pips, etc. You don’t have the time to count the exact pips before every move, and using your knowledge of the pip count is not an exact science anyway. However, after you’ve done it many times you should be able to get a feel for who is leading in the pip race fairly quickly. Knowing who is ahead is the vital information that you need; don’t worry too much about the overall totals.

The more you play, the easier you’ll find it to calculate the pip totals for you and your opponent simply by glancing at the board. This will also help you get a feel for how far advanced the game is. If you spend all your time playing against a computer, which calculates and displays the pip count for you, you won’t learn to count the pips for yourself and you’ll be at a disadvantage when you play a real person on a conventional board. So get out there and play some real games once in a while!

Can you make a quick calculation of who has more pips to go?

Can you make a quick calculation of who has more pips to go?

At first glance, the board shown seems to show a position where both players are on almost equal pips. However, if you look a little closer you’ll see that Black has a pip count of 139, while White has 153 pips to go, so Black has a pip count lead of 14 over you. Additionally, Black has thrown a 4+4 which gives him another 16 pips and he isn’t blocked so he can legitimately move all of his 4-moves (he should move 2 checkers from his midpoint to his 5 point) . At the end of his throw he will be 30 pips ahead, at 123 against your 153. That’s a big lead, big enough to change the way you should now think of how to approach this game strategically.

Becoming familiar with the pip count and knowing, at all times, whether you are up or down in the race will help you make better decisions. Sometimes you will be so far behind in the race to finish that your best chance of winning is to slow the race down by trying to delay your opponent. Knowing the approximate pip count will also help you make better decisions about doubling. You should always consider the pip count in your strategy for backgammon, no matter what stage you are at. Even the most experienced players sometimes forget (or don’t know) that there is no position in backgammon where the pip count isn’t a factor.

Again, this is why mathematicians excel at backgammon. Art Benjamin is a Math Professor at Harvey Mudd College. He is the author of several books on mathematics and has appeared on many television shows, entertaining audiences all over the world with his MathMagic show (he calls himself a mathemagician!) One year he decided to play in a lot of backgammon tournaments and ended up No. 1 on the U.S. tour!

Maths whizz, Art Benjamin

Maths whizz, Art Benjamin

The average value of a throw from two dice is 8.2, which is probably higher than you might have assumed. This is because doubles (where you get to make each move of the dice twice) push up the average. Remembering this can help you assess the game, especially in a race.

If you have an average of 8.2 moves on each throw, you need (on average) about 20 throws to make up your 167 pips to finish. Of course, a game is highly unlikely to go so smoothly that you are never hit (which means clocking up more pips), or blocked from moving, but it’s still useful information.

Many players drift through their matches without stopping to think because they assume that the direct route to getting round (i.e. in around 20 moves) is quite likely; but this isn’t the case. Players with this mindset try to make every move as safely as possible, avoiding leaving blots and taking risks, only to discover that this ‘safe’ strategy carries more risk than they’d bargained for. If they’ve safely stacked up their checkers on only a few points and haven’t made a number of good, strategic points to act as ‘stepping stones’ when they do need to move, they’ll find they become vulnerable at a more critical stage later on in the game. If your strategy for backgammon is always just to race to the end of the game faster than your opponent, backgammon will quickly become very frustrating because an opponent who has put a little more thought into their game can ruin your plans pretty much every time.

The actual number of possible moves in a backgammon game is rather mind-boggling. Let’s say you move first, then your opponent has his turn and then you move again. There are 21 different outcomes to any throw of the dice, apart from the first throw where there are only 15 because you cannot start with a double, so in just the first three moves of a game, there are 6,615 (15x21x21) different combinations. There are often more than 50 moves in a single game so the number of different combinations becomes astronomical and increases even more if the game is longer than that, due to players getting hit and blocked. Champion player and author, Walter Trice, estimated that there are over a trillion possible legal positions that can be attained on a backgammon board!

So it won’t work just to memorise some of these trillion moves. It’s much better to understand why a certain move is good and apply that thinking in different situations. Even if you learn the opening moves by heart, you still need to think the moves through every time and understand why you are making each choice.

With a better understanding of the game, you’ll be surprised how quickly assessing your next move becomes second nature to you. In life, sometimes we need to ignore the huge possibilities that lie before us and just live in the present, focusing on what we need to achieve in the immediate future. The same applies in backgammon. For the most part, you should just focus on your best move in the moment.

When you are using two dice, although there are 36 (6×6) possible dice throws, there are only 21 different outcomes because there are two different ways of throwing all the numbers except for doubles. For example, 3+4 counts as a different roll to 4+3. Because the different numbers could come up on either of the two dice, you are twice as likely to throw a combination of 4 and 3 than to throw a 4+4. Some people find this difficult to grasp at first.

I have listed all 36 possible throws below to help you estimate the odds of what you or your opponent might throw next.

 
Dice 1 Dice 2 Total pips visible on both dice (not doubles)
1. 1 and 1 2
2. 1 and 2 3
3. 2 and 1 3
4. 2 and 2 4
5. 3 and 1 4
6. 1 and 3 4
7. 3 and 2 5
8. 2 and 3 5
9. 4 and 1 5
10. 1 and 4 5
11. 3 and 3 6
12. 4 and 2 6
13. 2 and 4 6
14. 5 and 1 6
15. 1 and 5 6
16. 4 and 3 7
17. 3 and 4 7
18. 5 and 2 7
19. 2 and 5 7
20. 6 and 1 7
21. 1 and 6 7
22. 4 and 4 8
23. 5 and 3 8
24. 3 and 5 8
25. 6 and 2 8
26. 2 and 6 8
27. 6 and 3 9
28. 3 and 6 9
29. 5 and 4 9
30. 4 and 5 9
31. 5 and 5 10
32. 6 and 4 10
33. 4 and 6 10
34. 6 and 5 11
35. 5 and 6 11
36. 6 and 6 12

Now we’re familiar with the 36 different throws of the dice and the 21 different outcomes, let’s take a look at how likely you are to throw each number. This is going to come in handy because you’ll need to have a good idea of what chances you or your opponent have of throwing a particular number in many situations.

Let’s take the example that your opponent has two blots and you need either a total of 6 or a total of 10 to hit one of them, and to keep it simple let’s assume there is nothing blocking your way from either blot. You’re more likely to throw a total number between 1 and 6 than between 7 and 12 because it can be achieved with either one dice or two. Throwing a 10 can only be achieved with using the combination of moves on two dice, but throwing a 6 can be achieved with one dice or with a combination of two dice. You therefore have a lot more chances of throwing a 6.

1A good statistic to remember from the table of throws above is the number of chances of any particular number (not a combination) from 1 to 6 coming up with two dice. You have 11 in 36 chances of throwing any number from 1 to 6. For example, you can roll a 3 with any of the following 11 combinations: 3+1, 1+3, 3+2, 2+3, 3+3, 3+4, 4+3, 3+5, 5+3, 3+6 and 6+3. So for example if you have been hit and the only gap in your opponent’s home board is point number 3, you know that you have an 11 in 36 chance (just under a 1 in 3 chance) of re-entering with your next throw.

A good rule of thumb is that if you are trying to throw a number from 1 to 6, the higher the number the more chances you have because there are more ways to do it with two dice. For example, throwing the number 1 can only be done if at least one of your dice is a 1, so this only happens 11 out of every 36 times. But throwing a total of 5 can be done using four other combinations of both dice (3+2, 2+3, 4+1 and 1+4) as well as the 11 rolls that contain a 5, so you have a total of 15 out of 36 ways you can roll a total of 5 and move 5 spaces (assuming you’re not blocked from using any of the throws if you’re combining the dice).

To confirm this, take a quick look at the table listing the 36 throws. You can see that, in the case of the number 1, there are 11 times this single digit comes up, from 1 +1 all the way through to 6+1.

Here is a breakdown of the odds of throwing a number below 7, including combinations:

 
Dice 1 Dice 2 Total pips visible on both dice (not doubles)
1. 1 and 1 2
2. 1 and 2 3
3. 2 and 1 3
4. 2 and 2 4
5. 3 and 1 4
6. 1 and 3 4
7. 3 and 2 5
8. 2 and 3 5
9. 4 and 1 5
10. 1 and 4 5
11. 3 and 3 6
12. 4 and 2 6
13. 2 and 4 6
14. 5 and 1 6
15. 1 and 5 6
16. 4 and 3 7
17. 3 and 4 7
18. 5 and 2 7
19. 2 and 5 7
20. 6 and 1 7
21. 1 and 6 7
22. 4 and 4 8
23. 5 and 3 8
24. 3 and 5 8
25. 6 and 2 8
26. 2 and 6 8
27. 6 and 3 9
28. 3 and 6 9
29. 5 and 4 9
30. 4 and 5 9
31. 5 and 5 10
32. 6 and 4 10
33. 4 and 6 10
34. 6 and 5 11
35. 5 and 6 11
36. 6 and 6 12

Any old 6…. Odds of throwing a 6 with one or both dice: 17/36.

Any old 6…. Odds of throwing a 6 with one or both dice: 17/36.

Lucky 7… Odds of throwing a 7 with one or both dice: 6/36.

Lucky 7… Odds of throwing a 7 with one or both dice: 6/36.

There’s a big difference between the two, isn’t there? Without the magic of being able to throw your required number with a single dice, your chances get much smaller.

Most of us vaguely remember 7 is the most likely throw you can get in dice games such as Craps, because 7 is right in the middle of the range of numbers you could throw. But there are still only six ways to throw a 7. These are 5+2, 2+5, 4+3, 3+4, 6+1, and 1+6. That’s it.

To throw a 7 or above, we fall off a precipice in terms of our chances of getting a particular number with two dice, and it’s downhill from there in terms of how likely you are to throw any combination with a pair of dice. This is also very useful to know.

Here’s the rest of the list, showing the odds of throwing a number or combination of 7 and above:

 
Dice 1 Dice 2 Total pips visible on both dice (not doubles)
1. 1 and 1 2
2. 1 and 2 3
3. 2 and 1 3
4. 2 and 2 4
5. 3 and 1 4
6. 1 and 3 4
7. 3 and 2 5
8. 2 and 3 5
9. 4 and 1 5
10. 1 and 4 5
11. 3 and 3 6
12. 4 and 2 6
13. 2 and 4 6
14. 5 and 1 6
15. 1 and 5 6
16. 4 and 3 7
17. 3 and 4 7
18. 5 and 2 7
19. 2 and 5 7
20. 6 and 1 7
21. 1 and 6 7
22. 4 and 4 8
23. 5 and 3 8
24. 3 and 5 8
25. 6 and 2 8
26. 2 and 6 8
27. 6 and 3 9
28. 3 and 6 9
29. 5 and 4 9
30. 4 and 5 9
31. 5 and 5 10
32. 6 and 4 10
33. 4 and 6 10
34. 6 and 5 11
35. 5 and 6 11
36. 6 and 6 12

Remember this list includes doubles, so for example you have 3 chances of throwing a 12. You can do this with a 4+4, 3+3 or a 6+6.

There are gaps in the table because there are certain numbers which no possible combination of dice will deliver. For example, try as you might, you can never throw a 13!

You don’t have to learn this table but try at least to become familiar with it. Try to become at least on nodding terms with your chances of getting a particular number. It’s worth knowing, for example, that you have five times the probability of throwing a 5 than a 12. That’s a big difference and might well affect your thinking when making a decision in a particular situation.

Let’s look at two almost identical examples that show how having an understanding of this table can really make a big difference. In the first example, as shown below, Black has to throw a 7 in order for his backmarker to hit White’s blot on his 17-point. His chances of doing so are 6 in 36.

Black needs a 7 to hit; he has a 6 in 36 chance.

Black needs a 7 to hit; he has a 6 in 36 chance.

ut if your blot was just one point closer, and Black only needed to throw a 6 (as shown in the slightly different example below), Black’s chances leap up to 17 in 36, which is a big difference. Black is almost 3 times more likely to be able to hit the White blot in this second example just because your checker is now one point nearer to his backmarkers.

Here Black needs a 6 to hit and therefore has a much better chance (17/36)

Here Black needs a 6 to hit and therefore has a much better chance (17/36)

When you leave a blot that could be hit by your opponent, we call it leaving a shot. Always try to leave a long shot (preferably more than 6 spaces away) rather than a short shot as you can dramatically reduce your opponent’s chances of hitting it.

I played in a tournament recently where at one point in the game I had a checker on the bar because I’d just been hit. I needed to throw any number except for a 5 or a 6 (because my opponent had created a 2-point-prime on his 5-point and 6-point) in order for my checker to re-enter the board. At my first attempt, I threw 6+6. I could not get on. On my next turn I threw a 6+5; I still couldn’t re-enter. On my third attempt I was thwarted again, by throwing another 6+6. I failed yet again to re-enter. What are the odds of that? Well, they were a little under 730 to 1, so I was feeling rather hard done by!

However, later on in the same game my luck became as outrageously good as it had been bad and the game turned around in my favour. Finally I found myself needing to throw a double to win not just the game but also a rare backgammon and, sure enough, I managed to do so.

In the same game I had experienced unbelievably bad luck followed by extremely good luck. Backgammon is the only game of skill where your luck can fluctuate so dramatically. One thing I do believe (and every good player needs to believe in order not to go crazy playing this game) is that in the long run, luck evens out, and the better you play the ‘luckier’ you will be overall!