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Author Topic: How Many Set-Ups Are There?  (Read 2836 times)

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Dingan

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How Many Set-Ups Are There?
« on: July 29, 2015, 10:19:33 am »
+1

One of the reasons why I love Dominion so much is that you never play the same game twice.  And I also like discrete math.  So to that...

How many possible set-ups of Dominion are there?  I use "set-up" instead of "Kingdom" here because I want to include non-Kingdom cards -- Colonies, Shelters, the Black Market deck, etc.  Lets not include Adventures (so everything up to and including Guilds).  A lower bound would be the number of possible Kingdoms = number of available Kingdom cards choose 10 = 205 choose 10 ~ 3 x 10 ^ 16.

But then there are more things to consider:
  • Colonies/Platinums
  • Shelters
  • The cards in the BM deck (lets say the cards matter, but the order in which they're put there doesn't)
  • The Bane/11th card
  • What ruins are in ruins pile (Is it always 2 of each, or is it random, or what?  If it's always the same, then it wouldn't matter.)
  • etc.
Lets assume 2-players (so always 10 curses, 10 ruins, 8 per green pile, etc.).

Follow-up question: how long would it take to play all such games if every person in the world played Dominion all the time (1) assuming no set-up repeats, and (2) assuming repeats?  (Make whatever assumptions you'd need.)
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Spellbound

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Re: How Many Set-Ups Are There?
« Reply #1 on: July 29, 2015, 11:04:56 am »
+1

This is not precise. Assuming we have Platinum, Shelters and Events in account, and BM deck has every cards not in Kingdom in it.

n = numbers of every existing cards, ignoring YW
e = numbers of events
y = numbers of bane cards

(2^(e+2))[(n!)/(n-10)! + y(n!)/(n-9)!]
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GendoIkari

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Re: How Many Set-Ups Are There?
« Reply #2 on: July 29, 2015, 11:53:51 am »
0

A few things that make the number way larger than you might think at first:

- If the game has Ruins, then you have a different setup depending on which Ruins are in the Ruins pile. This would also depend on the number of players, so you would have to decide if you consider a setup for a 2 player game to be a different setup than a 4 player game. Not only is it which Ruins are chosen, but the order that the Ruins are in is a different game.

- The order that the Knights are in make for a different game, so games with Knights have 10! possible ways of having those Knights.

- The order of the Black Market deck makes for a different game. Since you can fill the Black Market deck however you want, not only do you need 2^(every card not in the Kingdom) ways to fill the Black Market deck, but then you have to multiply each of those possibilities by the number of cards in the deck factorial.

- If you do include Adventures, then remember that the limit of 2 events is only a suggestion, you can play with 2^20 possible combinations of events (including the no events option).

- Something that would actually lower the number of setups: by the official rules, you can't have Platinum/Colonies unless you have at least 1 Prosperity card. Similar for Shelters and Dark Ages.
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Dingan

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Re: How Many Set-Ups Are There?
« Reply #3 on: July 29, 2015, 12:05:06 pm »
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- If the game has Ruins, then you have a different setup depending on which Ruins are in the Ruins pile. This would also depend on the number of players, so you would have to decide if you consider a setup for a 2 player game to be a different setup than a 4 player game. Not only is it which Ruins are chosen, but the order that the Ruins are in is a different game.

- The order that the Knights are in make for a different game, so games with Knights have 10! possible ways of having those Knights.

- The order of the Black Market deck makes for a different game. Since you can fill the Black Market deck however you want, not only do you need 2^(every card not in the Kingdom) ways to fill the Black Market deck, but then you have to multiply each of those possibilities by the number of cards in the deck factorial.
I mentioned it's a 2P game, and that order of stuff (ruins, Knights, BM deck) doesn't matter, only which cards are there.  So Knights are basically a normal supply pile.  But ya, if any number of players and order mattered, the number would greatly increase.

- If you do include Adventures, then remember that the limit of 2 events is only a suggestion, you can play with 2^20 possible combinations of events (including the no events option).
Feel free to do that, but I mentioned no Adventures.

- Something that would actually lower the number of setups: by the official rules, you can't have Platinum/Colonies unless you have at least 1 Prosperity card. Similar for Shelters and Dark Ages.
Good point here, didn't think of that.  Definitely complicates things.  I wonder, do these things have to be in the supply, or can they be in the BM deck (sort of like how we can have Prizes when Tournament is in BM deck)?
« Last Edit: July 29, 2015, 12:06:49 pm by Dingan »
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GendoIkari

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Re: How Many Set-Ups Are There?
« Reply #4 on: July 29, 2015, 03:21:18 pm »
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- Something that would actually lower the number of setups: by the official rules, you can't have Platinum/Colonies unless you have at least 1 Prosperity card. Similar for Shelters and Dark Ages.
Good point here, didn't think of that.  Definitely complicates things.  I wonder, do these things have to be in the supply, or can they be in the BM deck (sort of like how we can have Prizes when Tournament is in BM deck)?

It's not like Prizes/Bane/Baker/etc; the card needs to be actually in the Kingdom. The rules state that you should randomly determine if you use Platinum with the same odds as the percentage of Kingdom cards from Prosperity; in other words pick 1 Kingdom pile at random, and if it is from Prosperity, use Platinum. So in addition to only being allowed to use Platinum when you have at least 1 Prosperity card, you aren't allowed to NOT use Platinum when you have a Kingdom of only Prosperity cards. Same from Dark Ages/Shelters.
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