One of my favorite board games is Settlers of Catan. I encourage all of you to check it out. It is a great game because it is a combination of luck and strategy and it is different each time you play. I’ve been on a big Settlers kick lately, because I’ve downloaded a version for my iPod Touch.
This post will probably be interesting only to those people who know the rules to the game (sorry), but I will give a brief explanation of the relavant rules so that anyone can follow the discussion.
A couple of years ago I asked myself the following two questions about Settlers of Catan.
- What are the most valuable intersections? This information is most important to know at the beginning of the game when placing the first two settlements.
- Where should I place the robber to do the most damage?
I just found my notes that I wrote, so I thought I’d share them here. The mathematical analysis is extremely simple, but is kind of fun and maybe a little useful to know.
The most valuable corners
At the start of the game the hexagonal game pieces are put in place randomly (as shown in the picture above, which you can click on to enlarge) and the round chips are distributed according the the rules. Each chip has a number 2 through 12 (except 7) on it. The players take turns placing settlements on the corners where three hexagons meet. Once each player has two settlements, play begins.
During each person’s turn a pair of dice are rolled and anyone with a settlement adjacent to a hex with that number on it gets a resource card of that type. If the player has upgraded a settlement to a city and it is adjacent to the hex, he or she gets 2 resource cards. For now I’ll ignore what happens when a 7 is rolled.
Thus it is good to put your settlement next to a 6 or an 8, 5 and 9 are good too, however the hexes with 2′s and 12′s are not worth much. You may put your settlement along the coastline or the desert, but the downside is that it would be adjacent to only 1 or 2 resource hexes.
The round chips not only have numbers, they also have dots on them. The dots give you an easy way to determine which are the valuable hexes. They are assigned as follows:
•: 2, 12
••: 3, 11
•••: 4, 10
••••: 5, 9
•••••: 6, 8
My question is: how do we look at a board and determine the most valuable intersections? By valuable I mean greatest likelihood of yielding a resource card when the dice is rolled. I’m ignoring the other strategies: settling on ports, spreading out your settlements, getting a broad range of resources, etc.
I was surprised to discover that the simple answer is the correct answer: add up the dots on the adjacent hexes. The intersection with the largest dot-sum is the most valuable.
The reason this is true is that if a chip has n dots, then the chance of rolling the number on that chip is precisely n/36. For example, there are 3 ways to roll a 10 (4 & 6, 5 & 5, or 6 & 4) and there are 36 possible rolls, thus the chance of rolling a 10 is 3/ 36. Indeed, the 10 chip has 3 dots on it.
Now suppose your settlement is on a 6/4/11 intersection (which corresponds to dots •••••/•••/••, or as I will write from here onward, 5-3-2). Then the chance that this settlement will pay off is .
Before answering the question I must point out that some combinations are impossible. You can’t have two adjacent hexes with the same (dice roll) number on them. You can’t have two adjacent •••••’s (or two adjacent •’s). You cannot have three adjacent hexes with the same number of dots. Thus, here are the most valuable intersections (of course, depending on how the board is set up, not all of these configurations may exist).
Probability 13/36: 5-4-4
Probability 12/36 5-4-3
Probability 11/36: 5-4-2, 5-3-3, 4-4-3
Probability 10/36: 5-4-1, 5-3-2, 4-4-2, 4-3-3
Probability 9/36: 5-3-1, 4-4-1, 4-3-2, 5-4 (on the coast or desert)
One consequence of this analysis is that you should not feel obliged to place every settlement adjacent to a 6 or an 8. There may more valuable corners elsewhere. For example, looking at the intersections in foreground of the picture above, the 8/5/10 and 8/5/4 (5-4-3) corners are the best, the 5/9/10 (4-4-3) is next best, and the 8/4/3 (5-3-2) comes after that.
The best use of the robber
The little black guy in the picture above is the robber (or bandit). He temporarily kills any tile upon which he sits (rolling a 7 allows you to move the robber). Obviously, you want to place it on a hex that has none of your settlements or cities and does as much damage to your opponents as possible.
It is important to note that properties (settlements and cities) cannot be placed on adjacent intersections, so you can have at most 3 on each hex.
My question is: where should you put the robber to do maximum damage to the other players? Again, I’m simplifying the analysis. Often you want to harm one player more than another because he or she is about to win—I’ll ignore that.
By the expected damage I mean the expected number of resource cards “lost” when the dice are rolled. For example if there are two settlements and one city on an 8-hex and an 8 is rolled, the other players will not get the 4 resource cards (1+1+2) they would have gotten had the robber not been present. Since the probability of rolling an 8 is 5/36, the expected number of lost resource cards by placing the bandit on the 8 is 4(5/36)=20/36.
It is easy to create a table of expected damages. The values at the top of each column of the table are the number of settlements plus twice the number of cities on the hex (the number of resource cards that would be distributed if that number was rolled). Each entry in the table is the expected damage (in 36ths)—so the larger the number, the greater the damage.
In short: to compute the expected damage, add the number of settlements to twice the number of cities, then multiply this by the number of dots on the hex.
For example, a 6-hex (•••••) with three settlements has a lower expected damage (15/36) than a 9-hex (••••) with 2 settlements and a city (16/36).