# Catan Analysis: Understanding the Numbers

*Catan, *or *Settlers of Catan*, is one of the most owned games recent years. A flagship game of the board game renaissance, *Catan *by Klaus Teuber has the players playing the parts of groups trying to settle on the mythical island of Catan. To settle successfully, they must deal with five different resources – brick, ore, sheep, grain, and wood.

The game is based around a series of randomly placed hexagonal tiles. These denote the five different resources (along with a desert) for the players to collect during the game. Each resource tile gets a number placed on it from 2 to 12 and the player collects that resource if the number is rolled.

The mathematics seem simple, and probability plays a vital role in the game strategy. It can make or break a player. The game even lends a hand with a probability score along the bottom of each number giving some idea as to how common that number is. How much more complicated can the math behind *Catan *get?

Well now, for the sake of today’s article – let’s take a look at some *Catan *analysis and see if we can make up our minds.

## Rolling Big – Numbers and Settlers of Catan

Let’s get mathematical…mathematical…

Okay, I can now see why Olivia Newton John went with physical exercise and not thrilling conversations about probability.

In *Catan,* the players get to roll two dice as their basic action in the game. This randomises the resources coming out each turn, giving 36 possible combinations between 2 and 12 to choose from. The most common number that could come out is 7, which there is six ways of rolling (6/1, 5/2, 4/3, 3/4, 2/5, 1/6). This is followed closely by rolling a 6 and an 8 of which there are 5 combinations of each. A 6, for instance, can be rolled 1/5, 2/4, 3/3, 4/2, and 5/1.

Rolling a 5 or a 9 can be rolled 4 ways each. A 4 or a 10 can be rolled 3 ways, 3 or 11 are rolled 2 ways, and 2 or 12 only have one way they can be rolled – 1/1 and 6/6 respectively.

So, that’s settled then, is it not? It is best to base your strategies on the numbers closer to the middle of the spectrum, as those are most likely to come out. Even though the 7 is the Robber in *Catan*, and not allocated to a resource, 6 and 8 are fairly safe bets. Is that not correct?

## Catan, Analysis, and the Art of Probability

Well, yes. It is. Statistically speaking, if there are more ways of rolling a 7 then the seven is the most likely number to come up. This is the mean of the mathematics. Coincidentally, if laid out 36 dice showing the 36 different rolls, then rolling a 7 is also the median of the numbers.

So, that makes rolling a 7 the single most likely number, which in *Catan *is the robber.

What that doesn’t do is automatically make 7 the most likely result to come up. Firstly, only 6/36 (1/6) rolls are likely to be a 7. Secondly, in order to take probability really seriously, we need to look at how many times we are rolling the dice.

Probability with dice takes into account a period of time/rolls, with the idea that the dice will average out over a large number of rolls. If you roll the two dice 100 times then the odds are that 7 will be the most common number. If not then it will be if you roll the dice 200 times, or even 1000 times. Over time, over a period of rolls, the numbers will even out so that 7 is the most common number rolled.

That being said, the average game of *Catan* apparently lasts for 80 rolls. That’s 20 turns in a four-player game, or 20 rolls each. The odds are that isn’t enough rolls for the probability to even out. If it does even out then it certainly wouldn’t be a noticeable difference. The important thing to keep in mind, is that even though there is a 1/6 chance it does not mean that 1/6 rolls will be a 7.

What this means is, in a game of *Catan*, it is likely that the difference comes down to the difference between the dice rolls. In the short term, for instance, there is probably little difference between rolling a 7 (6 different options) and a 6 (5 different options). In a game, 6 may even be the most rolled number; however, there is almost certainly going to be a huge difference between rolling a 7 (6 different options) and a 12 (1 option).

### An Experiment with Three Sets of 2D6

The problem with games like *Catan*, when working out the probability of rolls, is that each game of *Catan *exists as its own singular event. If you were to add all the results of all the games of *Catan* together then in all probability, the most common number should be 7 when rolling 2D6.

So, I decided to do an experiment. For the past 45 minutes, I have been rolling three sets of 2D6 and logging the results whilst watching Fargo Season 3 (that part isn’t relevant, I’m just adding some context). The results of the three sets of rolls are interesting, and I have graphed them below.

As you can see, the very first set of rolls did not conform to what we would expect from the averages. The number 7 only came up 6 times, which is weirdly the same as the number 12. That being said, it is not unusual for dice to be weighted in some way, so I decided to redo the experiment with two other sets of dice.

The second set of dice rolled in a way that would correlate more with what we would expect from 2D6. The number 7 was the most common.

The third and final set rolled making the number 8 the most common number, which isn’t surprising given the small number of rolls.

What is important though, looking at the graph, is that, although the rolls didn’t come out as expected (or even uniform), they do adhere to a curve. Where there isn’t much between 6,7, and 8, there is a difference between those and more extreme numbers on the chart.

When we add the three rolls together to simulate 240 rolls we see more of a pattern emerging.

Where the results are still not quite as we would expect, they are far more uniform. They do create a bell curve, more-or-less, with the peak being around 6,7, and 8.

## So What Does This Mean For Catan Strategy?

Of course, all of this *Catan *analysis isn’t for nothing. The question is, what does it mean for creating a strong *Catan* strategy? Well, what it means is a few things in regards to maximising your game.

**Firstly**, if a resource is on low probability numbers (so, for instance, Brick may be on a 2, a 4, and an 11) then it may be worth putting more emphasis on those numbers. By straddling two or three low numbers then you have just as much probability of getting a resource come up during an 80 roll game then you do if you place on more likely numbers like 6 and 8. This assumes you can straddle those numbers at once. Of course, it isn’t as efficient, but it will allow for you to monopolise on a resource later on, even if you do have a slower start to the game.

**Secondly**, the most commonly needed resources near the start of the game are Brick and Wood, so getting on those near the start and monopolising them (no matter what their numbers are) is a good strategy. If they are on common numbers then this isn’t always possible, in which case Grain is the best mid-game resource to go for. It is used for everything bar building a road. Finally, if you want to control the endgame then control the Ore. Ore is needed to city-up and buy development cards, something which is needed to be done in order to win most games.

Without Ore, a player can never get more than 7 points.

**Thirdly**, the opposite is also true. If something is on the more probable numbers then it isn’t likely to be a huge factor in the negotiation side of the game as everyone is likely to have it. This means it will be harder for someone to monopolise on that resource, making it a more common (and thus cheap) trade resource later on.

**Fourthly**, if a likely number is right next to a port then go for it. This is basic *Catan *strategy but it is always worth keeping in mind.

## Catan Analysis: Conclusion

So, there we have it. A brief breakdown into the numbers behind *Catan*, the probability/what it means, and which are the best way to play them. Basically, they aren’t as simple as they may appear, so it isn’t always the best strategy to go for the 6 and 8 and call it quits. There are plenty of ways to play the system to get a spread of what you need.

Through writing this we also had a load of other *Catan* article ideas, so expect a few more soon.

Now, over to you. What are your thoughts on the mathematics behind *Catan*? Let me know in the comments below.

Some thoughts:

I agree that with just 20 rolls the difference between the number of likely 6, 7, and 8s rolled will be small. The bell curve really starts to look more “normal” once you reach around 50 rolls.

The insight that placement near rarer goods can increase your chances is a useful one, as is a reminder that the relative worth of the goods shifts through the game.

My last thought is rolling those dice seems like the slow road to simulation. It looks like you are creating charts using excel or similar, I suggest having a look at the random number function, then you could smash out the data for a couple of hundred rows in a few seconds.

I do enjoy seeing a bit of data in a game analysis, thanks for posting.

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So, I actually tried this earlier today and yeah, you’re right. It is a much better way of simulating rolls.

I was doing it as two 1-6 Random Number generators and then =sum(_,_) them together to simulate 2D6? Is there a simpler way? How would you do it?

I actually kept getting 6.8 or something along those lines for the result. I’m wary of using a =roundup(_,0) or =rounddown(_,0) function as I don’t want to round the wrong way. Any thoughts?

Sorry, a complete bombardment of Excel questions for you there 😛

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just use round(,0), forcing it to round one way or the other might cause a little bias. I also just use + or sum. Glad you are finding this a useful tool. Cheers

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So, I actually took your suggestions and managed to simulate 10,000 games of Catan earlier this week. Really helpful advice, so thank you 🙂

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welcome to the world of monte carlo simulation 🙂

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Wow, I am envious of your ability to do a post like this.

Well done!

I still have only played Catan once. I used to play it on Xbox some, and I do have the iOS app but haven’t opened it in eons.

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I wouldn’t be envious. I just have too much free time 😛

Means a lot though mate, so thank you.

Also – play Catan! You won’t regret it! It’s amazing!

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I love stats, I love Settlers, and this was a fantastic merging of the two! =-) It even included a nice bell curve and some implied standard deviations. =-)

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Ahh sorry, this got filtered into my spam for some reason so I’ve only just seen the comment! I’m glad you like it 🙂

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Nice article! We use similar placement strategies in Catan, looking at most commonly rolled numbers, but also key resources. I had never thought about monopolizing ore at the start though. That would be hard to build up from, but I might have to keep that one in mind, depending on where the ore tiles land on the board. I agree with dave2718, you can do much larger simulations with random number functions. I’ve done that with RPG systems in the past.

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Oops, I forgot to mention that we also tend to watch out for wonky dice. Seems like you might have had one in your first batch of rolls.

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Yes, maybe. And monetising ore is a risky strategy, but it can pay off 🙂

I tried the random number generator earlier on today and yeah, you’re both right. It is way easier.

One of the things I learned to do for my job a little while back is to build a Python script for randomly putting together pieces of information. Using that same methodology it might be possible to create a truly random dice generator that works out all the math and averages of doing 80+ rolls. That way I can say goodbye to rolling random, potentially dodgy, dice 🙂

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Hey, nice job! There are online random number generators. Not sure if that’s what you used. Or yes, it’s simple to script. I once started coding a combat simulator so I could test various monsters for a game I was working on. My scope was too big though, and I didn’t end up completing it. Maybe someday.

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That sounds incredibly complicated but really cool! Your list of variables must have been phenomenal. How far did you get with the code? A whole wargame simulator is one heck of a task!

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Yea, it’s basically the same as any rpg computer game, except I would be automating both sides. I built a simple one that was 1 hero vs 1 monster, with most variables hard set. But trying to add more combatants for really complicated. I also wanted to make it generic enough that I could use it for any RPG system. Like I said, way too ambitious of a project!

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Yeah, I can see that getting complicated. I’ve tried doing that with Warhammer demons, working out who would beat what. Those only have a few variations and it was still a nightmare.

I imagine that was simple when it was A vs A – but adding in an additional variable it would turn to A vs A, A vs B, B vs A, B vs B. A third variable would be A vs A, A vs B, B vs A, B vs B, A vs C, C vs A, C vs C, C vs B, B vs C, A vs B vs C etc.

I can definitely see how that would get out of hand quickly.

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Yep! Big hang up was just initiative order. Adding combatants in and out of the combat cycle. Who is on which side, etc. Maybe someday I’ll try again, with a much simpler version.

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Dude – sounds incredible. Maybe one day 🙂

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Nice write up. We have moved on to the card deck for number generation as it still has some randomness as you bury 5 cards each “year” and there are bonus results the cards call for as well. We have found they make the game more even than the dice do.

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That’s a really interesting way of doing it. Do you prefer cards over dice then?

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Yes, makes the game more fun and incentizes getting largest army, longest road, most harbors, while also leveling the playing field against a runaway winner. Have you tried the cards?

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No, I haven’t. Are they specialist cards then?

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Yes, and I think we got them in one of the expansions, maybe traders? There is a reshuffle card “new year”that gets 5 random cards buried underneath it. The number of each potential die roll is equal to the number of dots on the cards, so there is one 2 and 12, and six 6’s and 6 8’s with all the numbers getting the right possible frequencies. I looked on the Catan website and did not see it, but it is here on ebay https://www.ebay.com/itm/NEW-Settlers-Of-Catan-44-Event-Cards-Pack-FREE-SHIPPING/122080424652?hash=item1c6c8f72cc:g:kxMAAOSwawpXrKR5

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That’s really cool. I’ve never played Traders but will definitely look into the cards. Thanks for the link 🙂

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