How Many Decks? Part 6: The Filth
The most complex mess of combinations yet.
Happy Halloween! I wasn't planning this, but it is an especially appropriate day to extend this series of "How Many Decks?" articles to consider the Filth. (I just had a mental picture of a group of trick-or-treaters dressed up as various members of the Filth faction. Now that's a scary thought ...)
Anyway, the Filth are the most complicated faction to count for three reasons. First, they have a whole new type of Unit, the Mutation, in addition to the Commons and Champions that everyone else has. Second, the number of Mutations in a deck can vary: you can have as few or as many Mutations as you want, so the number can range from 0 to the total number of available Mutations. Third, a Mutation can take the place of either a Common or a Champion, so even if we decide on 7 Mutations, say, those Mutations can take up 0, 1, 2 or 3 Champion slots, each of which has its own numbers of decks we could build.
Before we roll up our sleeves and get to work, let's point out that while the Filth will probably have far more decks possible than the other factions, most of them will be unusable in real play, The Filth need a balance between Commons to mutate and Mutations to turn them in to. Too many on either side and the deck will be a flop. But for math geeks like me there is a fascination in just seeing what's possible; so let's start counting!
The Basic Filth
Fortunately, when we look at the starter Filth deck we find a simplification to our task, namely that Champions and Mutations are treated just the same as far as counting decks is concerned. For each of them there is only 1 copy allowed in the deck, so we don't have to vary their numbers separately. We just vary the total of Champs+Mutes from the 9 in the deck, down to the 3 minimum Champ slots, as seen in Fig. 2.
Let's say we want a deck with a total of 7 Champs and Mutes -- row 3 in Figure 2 -- and run the numbers. As we discussed in Part 1, the number of ways of choosing 7 of them out of 9 choices is
C( 9,7 ) = 9! / 7!2! = 9*8 / 2*1 = 36
Then in Part 2 we learned that to choose the remaining 12 Commons out of 2 piles is
C( 12+(2-1), (2-1) ) = C( 13, 1 ) = 13/1 = 13
Then we subtract out the combinations with too many Commons. Both Zealots and Cultists start with 1 on the Battlefield, so if we choose 10 more to guarantee too many, that leaves 2 free slots:
C( 2+(2-1), (2-1) ) = C( 3, 1 ) = 3/1 = 3
13 - 3 - 3 = 7
For a total of 36*7 = 252 combinations.
Here is what we get for each of the rows in Figure 2:
|Champs + Mutations||Commons||Combinations|
2223 is a lot more than the 60 to 73 that the other factions had in Part 2!
Hopefully you will be receiving Saella's Precision soon, if you don't have it already. It adds several very interesting new Mutations to supplement the Demagogue's toolbox, as well as 2 nice new Champions. Hmm, not actually "nice", but you know what I mean.
These reinforcements add good game play options, but combinatorially they just increase the numbers: 3 Champions rather than 1, 13 Mutations rather than 8, and 3 kinds of Commons rather than 2. (Oh right, the Anointed are useful, too!) So the range of Champs + Mutes can go from 3 up to 13 now, and we can still lump them together since we it is not yet possible to grab more than 3 Filth Champions. Running the numbers again, we get:
|Champs + Mutations||Commons||Unlimited Combinations||One of Each Deck Combinations|
Column 3 is for combinations with the numbers of Commons limited only by the rules limit of 10, as discussed in Part 3; column has Units limited to just what you get if you use only 1 copy of each deck, as discussed in Part 4. You will notice that there are 0 such combinations: there are 5 uncommitted Cultists and Zealots, and 5 Anointed, so we can't get more than 15. (It's a comfort when the formulas add up to what you expect!)
Again, the totals are a lot more than the maximums of 66,960 and 14,724 that we saw with most other factions, though not nearly as big as the 7,868,950,752 and 6,344,464,464 that the Mercenaries had.
Filth with Mercenaries
When we add Mercenaries to the Filth, I must confess that I am at the end of my combinatorial rope. We have these constraints:
Total Units: ChampsFL + ChampsMR + MutationsFL + CommonsFL + CommonsMR = 21
Total Champions: ChampsFL + ChampsMR <= 3
Setup Commons: CommonsFL >= 2
Total Commons: CommonsFL + CommonsMR <= 18
Total Mutations: MutationsFL <= 13
Total Mercenaries: ChampsMR + CommonsMR <= 6
Most of these constraints are upper or lower bounds, not exact numbers, so the number of each kind of unit can range a lot. Also, the constraints are multi-dimensional: the number of Merc Champs, for example, is subject to both the total of Mercenaries and the total of Champions -- and we cannot simply combine the Champions and Mutations any more since now we can choose more than 3 Champions. Because of these multiple, interacting constraints, I could think of no way to simplify the counting short of listing every possible set of values for the types of units listed in the constraints.
The number of distinct numbers of types of units turned out to be 760! Fortunately, it helped to be able to copy, paste, and adapt subportions of the list rather than doing each entry by hand.
For each entry I used the same sorts of formulas that we have discussed, so I will cut to the bottom line. The total number of Filth decks that could be built while using Mercenaries is ...
... drum roll ...
... 7,834,397,445,264 for one-of-each-deck combinations, and 11,668,289,075,847 for unlimited combinations!
Almost. Twelve. Trillion.
Breaking it Down
That's such a big number that I wanted to see how the combinations broke down by the types of Units involved. The results are seen in Figure 3. (In each chart the blue line is for unlimited combinations, the red line for one-of-each-deck combos.)
In the top two charts we see that the number of combinations goes up rapidly as the number of Champions increases, and as the number of Mercenaries increases. In the bottom two charts we see that there is an optimum number of Commons, and of Mutations: their curves are similar to the famous Bell Curve, dropping down to near zero on either side. If the Champions and Mercenaries were allowed to range high enough, they would probably also hit a maximum and come back down as well.
The charts for Commons and Mutations look almost like mirror images of each other, which makes sense since their totals mostly trade off with each other. We'll end this installment with a look at a combined chart of unlimited combinations when both Commons and Mutations are varied:
We see a narrow bell-type curve aligned along the diagonal line that shows the inverse relationship between Commons and Mutations, namely that as one increases the other must decrease. This is just what we would have expected -- if we had been smart enough to figure out what to expect.
This isn't quite the end of the series. There are some last thoughts I'll want to work out before bringing it to a close. Until then!