There’s more math to it than you might think
Settlers of Catan is a highly strategic game. You have to balance your timing of buying properties and expanding your reach. You have to decide when it’s a good idea to trade and when you’re better off keeping your prized resources to yourself. In a similar vein, every time you roll a “7” or draw a robber, you have to determine what kind of player you want to be—do you want someone to be out for revenge or are you going to play nice?
Here are some math-based tips to help you as you play:
1. Know exactly what the dots mean when deciding where to place your settlements
As I’m sure you’ve noticed, the dots on the resource tiles indicate how likely that number is to be rolled. An 8 (5 dots) is far more likely to be rolled than a 2 (1 dot), simply because of the number of ways two dice can be rolled to make the numbers. For example, an 8 could come up as 2 and 6, 6 and 2, 3 and 5, 5 and 3 or 4 and 4, whereas the only way to get a 2 is by rolling double ones.
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Using this logic, you can figure out that 8 is five times more likely to be rolled than 2. However, the relative worth of numbers gets smaller as the number of dots goes up. The likelihood of a 2-dot number being rolled is double that of a 1-dot number, but there is only a 25% increase between a 4-dot and 5-dot number.
2. Know how many of what resources you need from the get-go.
The number of settlements, roads and cities you can build is limited so you only need a certain number of each resource over the course of the whole game even if you build them all.
Roads (1 brick and 1 wood each): For all 11 roads, you need 11 brick and 11 wood.
Settlements (1 wood, 1 brick, 1 sheep, 1 wheat): You start with two free settlements, so you don’t need to count the resources for them. However, once you start upgrading your settlements to cities you get some back. So, if you upgrade the original two, you can build 5 settlements, which will give you 9 points. You won’t be able to build any more cities than that because as soon as you upgrade one of your other cities to a settlement to get a piece back, you’ll go over the 10 points needed to win the game. So in total you need 5 wood, 5 brick, 5 sheep and 5 wheat.
Cities (2 wheat and 3 ore): For 3 cities, the maximum number you can build, you need 8 wheat and 12 ore.
So, in total, you need a maximum of 16 brick, 16 wood, 5 sheep, 13 wheat and 12 ore.
Of course, this doesn’t take into account trading (with other players or through a port) or buying development cards, but it does give some insight into the relative value of resources.
3. Have a plan
Going for either the longest army or the longest road is a typical strategy. Going for both takes a lot of time and resources, and they both give you two points at the end of the game, so how can you pick which one is best?
Based on the likelihood of drawing a knight as a development card (56%), you can figure out that it will take about 6 development cards to get the three knights you need to claim the largest army. That will cost you a total of 18 resource cards.
Getting the longest road takes at least 5 road segments — 10 resource cards — so it is cheaper than the largest army to acquire.
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Keeping both of these can be costly: about 12 resources to get two more knights and 4 resources to get two more road sections.
So, just based on resources, the longest road is both cheaper to acquire and to keep. On the other hand, it depends on what your opponents are going for. If you have to fight a lot to keep the longest road and nobody else is buying knights, you may need to change your strategy.
Whether or not you should use the robber comes down to game theory and knowing the other people you are playing with. You don’t necessarily get long-term gain by putting the robber on someone else’s territory if they retaliate. A good strategy to use would be tit-for-tat: be nice to those who are nice and build cooperation, and retaliate against those who steal from you. By doing so, you will teach your opponent that it isn’t worth placing the robber near you because you will get back at them next time you get the chance. However, the game relies heavily on trading so you don’t want to go unnecessarily making enemies.