# This board game blew my mind

Terra Mystica, a board game which has remained in Board Game Geek’s all-time top five for years, is like nothing I have ever played before.

The base game supports 2-5 players and includes 14 factions, each with a unique set of abilities.

As a game geek, I wondered how you play-test something with so many combinations.

This led to the question: How many faction combinations are there in the game?

For those who know binomial coefficients, calculating the answer might sound simple – but Terra Mystica throws a spanner in the works.

### When you choose a faction, you eliminate another

Terra Mystica has 14 factions, or, more accurately, 7 pairs of factions.

Each couple is a denoted by a colour, and when you choose one faction, you eliminate the other in the couple.

The implementation is simple – double-sided print. To choose a faction, players must take a board which has their faction’s features printed on one side.

The other faction in the couple is printed on the other side of the board.

The factions and their colours are:

2. Red: Chaos Magicians, Giants
3. Blue: Swarmlings, Mermaids
4. Grey: Dwarves, Engineers
5. Brown: Halflings, Cultists
6. Black: Alchemists, Darklings
7. Green: Auren, Witches

For the mathematics, this choose-one-eliminate-one mechanic complicates things somewhat.

### The maths

The first player to choose a faction has a selection of 7 faction boards, the next player 6, and so on.

Multiplied together, this gives you the number of permutations.

When you multiply incrementally-decreasing numbers, you get a factorial. There are seven factorial (7!) total permutations of the faction boards. This equals 5,040 permutations.

However, you don’t have seven players in a game. Four players, for example, results in 7 × 6 × 5 × 4.

To get rid of 3 × 2 × 1 in the original calculation, divide by 3!. This equals 840 permutations in a four-player game.

This is what the permutation calculation looks like written as a formula.

• n is the total number of faction boards
• k is the number of players

However, the number of permutations counts the same set of four factions multiple times, where factions are assigned to different players.

For example, this is one permutation:

• Player 2 — Mermaids
• Player 3 — Halflings
• Player 4 — Auren

Another permutation might have the same factions, just assigned to different players:

• Player 2 — Mermaids
• Player 3 — Auren
• Player 4 — Halflings

To eliminate this repetition and get the unique number of faction combinations, you divide by the number of permutations of players.

There are therefore 35 combinations of faction boards in a four-player game of Terra Mystica.

This is was the combination calculation looks like.

• n is the total number of faction boards
• k is the number of players

It is so important it even gets its own notation.

We can now calculate the unique combinations of factions in a four-player game.

To calculate the number of unique faction combinations in a four-player game, you start with 14 possible factions. One is eliminated and the next player has a choice of 12. The next has a selection of 10, and the fourth player can choose one of eight factions.

When you take out a factor of two from each of the terms in your calculation, it starts resembling the factorial we’ve been working with – multiplied by a power of two.

As a formula, you get:

• n is the total number of faction boards
• k is the number of players

You can make it even more general by using x for the number of sides per board.

Yes, physical boards can only be printed on two sides, but what if the player mat could be a cube?

Using this formula, the number of unique faction combinations per group of players are:

• 2 players: 84
• 3 players: 280
• 4 players: 560
• 5 players: 672

### Terra Mystica

Terra Mystica doesn’t come cheap in South Africa.

On Amazon, a new copy is \$61.90, excluding shipping and taxes.

Locally, you’re looking at around R1,200 from Dark Aura – ranging up to R1,600 from other retailers.

The game comes with lots of little bits, a pretty game board, and seven mathematically-gorgeous faction boards.

Thanks to Shaun for his help with the calculations.