Welcome to Part 2 of this reprint series, celebrating the imminent arrival of the final Reinforcement Packs for Summoner Wars, along with the reprint of out-of-stock items, and the first publication of Mice and Mystics!

 

For many people, one of the most enjoyable things about Summoner Wars is deck-building, and in this series of posts I am investigating the question: How many different decks can we build, for any particular faction?  In Part 1 we figured out how many ways there were of selecting 3 Champions for a customized deck for every faction.  Now it’s the Commons’ turn!

Fig. 1. Tundra Orc Commons in the Starter Set
Figure 1. Tundra Orc Commons in the Starter Set

As before, our initial examples will come from the Tundra Orcs, who have 3 kinds of Common Units to choose from the Starter Set: Fighters, Smashers and Shamans.  (We will leave the Chargers and Thwarters from the Reinforcement Pack for another time.)

 

Figuring out the number of combinations of Commons is a more complicated calculation than Champions for several reasons:

  • We need 18 Commons as opposed to 3 Champions.
  • There are multiple copies of each Common, while there is only 1 of each Champion.
  • The number of any kind of Common is limited to 10.
  • Several Commons are required for the initial setup.

So we will start the simplest scenario, and build from there.

 

Starter Set Commons Only

To begin, let’s go back in time to the release of the Summoner Wars Starter Set Phoenix Elves vs. Tundra Orcs, and let’s pretend that we have 2 copies of it.  (Well, at least some of us have to pretend.)  Now let’s see how many ways we can fill out our allotment of Common Units.  First we will set aside the 2 Fighters, 1 Smasher and 1 Shaman that we always need at the start of the game. That leaves us 14 more slots to fill, and 3 piles of Commons to choose from:

Fig. 2. Choosing the Tundra Orc Commons for the deck.
Figure 2. Choosing the Tundra Orc Commons for the deck.

Here’s a clever trick that is used make it easy to count the combinations.  We start at the first pile, draw as many cards from it as we want, then go to the second pile, draw what we want from it, and then go the third pile and finish our selections from there.  That’s so simple it doesn’t even seem tricky, does it?  But let’s make a chart of our choices, using ‘d’ for the act of drawing a card, and ‘>’ for the act of moving to the next pile.  So if we want to duplicate the Starter Set, we would draw 6 more Fighters, 4 more Smashers and 4 more Shamans:

dddddd > dddd > dddd

If we decided we hate the way Shamans always seem to miss, we could fill up on Fighters (8 more to reach the maximum of 10), and choose the rest as Smashers, who are Sluggish and easy to hit, but will hit more often than Shamans.  Notice that we shift to the Shaman pile but do not choose anything there:

dddddddd > dddddd >

 If we want to avoid Fighters we can start by skipping them and split the rest between Smashers and Shamans:

> ddddddd > ddddddd

And if we’re egalitarian we can make our deck have 6 total of each:

dddd > ddddd > ddddd

In each case we have 14 d’s because we choose 14 cards total, and we have 2 >’s because we have 3 piles, we start at the first, and we must shift piles 2 times to get to all of them. 

Here is where the trick comes in: in general, if we want to choose K cards from N piles, it’s the same as having K+(N-1) characters, and choosing K of them to be d’s.  This is the same as choosing (N-1) of them to be arrows.  As we saw in Part 1, this is a basic formula for combinations:

C( K+(N-1), K ) = C( K+(N-1), N-1 ) = (K+(N-1))! / K!(N-1)!

For the Tundra Orcs, we then get

C( 14+(3-1), 3-1 ) = C( 16, 2 ) = 16*15 / 2*1 = 120

 

Eliminating Illegal Choices

Now, not all 120 choices are allowed.  For example,

> > dddddddddddddd

Would give us 15 Shamans total, which is not good.  First, it’s probably unwise for gameplay.  Also, we don’t have 15 total Shamans in 2 copies of the Starter Set.  But even if we did, it’s forbidden by the rules, which stipulate a maximum of 10 copies of any Common Unit.  So what do we do?

The simplest thing to do is to count the number of illegal combinations of Commons, and subtract that from the number of total possible combinations. We’ll have to do that separately for each kind of Common.


Figure 3. Choosing too many Fighters -- at least 11.

In the figure above we see the starting Commons set aside, along with 9 more Fighters, making 11 total, which guarantees that no matter what else we choose this will be illegal.  So we have to choose 5 more Commons from the 3 piles:

C( 5+(3-1), (3-1) ) = C( 7, 2 ) = 7*6 / 2*1 = 21

There are 21 ways of selecting TO Commons that end up with too many Fighters.  (This includes choices that include more cards than most of us have, even in 2 Starter Sets, but that’s OK.  We’re counting theoretical possibilities.  As I said, we’re pretending.)

Fig. 4.  Choosing too many Smashers
Figure 4. Choosing too many Smashers -- at least 11.

The calculation for Smashers is different because there is only 1 Smasher on the board to start, vs. 2 Fighters.  So we start with 10 extra Smashers, therefore we only need to choose 4 more total Commons:

C( 4+(3-1), 2 ) = C( 6, 2 ) = 6*5 / 2*1 = 15

The numbers for the Shamans are the same, so there are also 15 ways of making TO decks with too many Shamans.  Therefore, the total number of legal combinations of TO Commons in the Starter Set is

120 – 21 – 15 – 15 = 69

 

Putting Them Together

Now we are ready to compute the total number of decks we can build with Starter Set material, since we know the numbers for both Commons and Champions.  There is a theorem in combinatorics that says that

If event P can happen in p different ways, and Q can happen in q different ways, then if P and Q are independent of each other — that is if any choice of one does not affect a choice in the other — then the number of different ways for (P and Q) to happen together is p*q.

This is good to know, since the choices of Champions and Commons are independent.  We need 3 Champions and 18 Commons, and they come from different sets of Units, so the choices have no effect on each other (mathematically at least — gameplay is a different question altogether).  Using unlimited numbers of Starter Set Units, we have 69 combinations of Commons, while of the Champions we saw that we have … well, one — only one way of choosing 3 Champions out of 3.  Multiplying, 1*69=69, so there are 69 different TO Starter Set decks.

Let’s not stop there.  Let’s see how many decks there are for each faction, based only on varying the number of each type of Common available in its starter deck.  This will depend only on the number of each type of Common that is in the starting setup:


Combos   Setup#   (Factions)

        73       1,1,1        (Swamp Orcs)

        69       2,1,1        (Tundra Orcs, Phoenix Elves, Fallen Kingdom, Mountain Vargath, Mercenaries)

        68       2,2,0        (Benders)

        65       2,2,1        (Guild Dwarves, Jungle Elves, Cloaks, Sand Goblins)

        64       3,1,1        (Vanguard, Deep Dwarves)

        60       3,2,1        (Cave Goblins, Shadow Elves)

          9       1,1           (Filth)

You will notice that the fewer Units on the battlefield to start, then the more units there are to choose from, and therefore more ways of building the deck.  The exception is the Filth, who only have 12 Commons of 2 kinds rather than 18 of 3 kinds.  This isn’t accurate for the Filth, because they can legally decide to exclude some or most of their Mutations and fill the deck with more Commons.  But that makes a more complicated calculation, so we’ll leave it until later.  We have done enough for this segment.

Next time, Reinforcement Commons!

 


Comments


MacBryce commented:

Nice!

Posted on 2012-10-02.

Shockma Ranyk commented:

And that is how you MATH.

Posted on 2012-10-02.

TheOne commented:

BOOM. Knowledge. You just got articled.

Posted on 2012-10-02.

worlds_enemy commented:

This helped me tremendously. Thank you.

Posted on 2012-10-03.

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