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[kilgoretrout103]

Theory stated in his excellent review of Dominion that there are roughly 2 quadrillion different Kingdom combinations. I assume he arrived at this by calculating 157 choose 10, which is about 1.8 quadrillion. That's a fairly good starting point, but it fails to take into account the presence of a Bane card, or whether a game has Colonies/Platinums.

In the Isotropic downtime this morning, I started thinking: exactly how many different Kingdoms are there? Two hours and a bunch of combinatorics later, I arrived at my answer: 1*10^24, or about 1 septillion! That's nine orders of magnitude higher than theory's estimate! Jesus.

Anyway, just thought I'd share my results from an interesting (if not strategically relevant) problem. I attached a doc showing my work. It presupposes a basic understanding of Dominion and discrete math, but if any other game-playing mathematicians out there want to check my work or comment on it, I'd love to hear from you!

EDIT: You need a recent copy of Word to open the document... I have a PDF, but it's slightly too big to post here. PM me with your email address and I can email you a copy.

EDIT2: Do a typo when I input my formula to Wolfram Alpha, I am revising my answer down to 1.23 x 10^16, or about 12 quadrillion.

[WanderingWinder]

Theory stated in his excellent review of Dominion that there are roughly 2 quadrillion different Kingdom combinations. I assume he arrived at this by calculating 157 choose 10, which is about 1.8 quadrillion. That's a fairly good starting point, but it fails to take into account the presence of a Bane card, or whether a game has Colonies/Platinums.

In the Isotropic downtime this morning, I started thinking: exactly how many different Kingdoms are there? Two hours and a bunch of combinatorics later, I arrived at my answer: 1*10^24, or about 1 septillion! That's nine orders of magnitude higher than theory's estimate! Jesus.

Anyway, just thought I'd share my results from an interesting (if not strategically relevant) problem. I attached a doc showing my work. It presupposes a basic understanding of Dominion and discrete math, but if any other game-playing mathematicians out there want to check my work or comment on it, I'd love to hear from you!

Oh man, you're not doing black market right. 30 cards in the BM? Where did you get that? You can have whatever you want in the BM, so long as it's not already in the kingdom.... Which of course means that your estimate is many, many, many orders of magnitude too small. Also, with the young witch, you choose the other kingdom cards BEFORE grabbing the bane, and yes, that makes a difference. Not to mention, why can't you have platinum and colony without any other prosperity cards? Sure, it's not recommended, but they also recommend that you have multiple alchemy cards if you have one....
Anyway, my point is that how many there are depends very much on what restrictions you put on, and if you just follow the rulebooks in the broadest possible sense... there's a LOT. Way too many to calculate - it's a very intricate thing, mostly because of black market.

[kilgoretrout103]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10. Also, on Isotropic, the Black Market deck is 30 randomly chosen cards, not all the extra cards.

Also, I am not counting differences in Black Market decks as different cases. The exercise is to determine how many possible KINGDOMS exist. The Kingdom and the Black Market deck are independent, except when Young Witch is in the Black Market deck, which is a case I study.

As for the order of when you choose the cards (that is, Bane first or the rest of the Kingdom first), I don't think that actually impacts the number of possible outcomes, so I think it's irrelevant. Could you explain to me how it might matter?

[werothegreat]

^ What he said.  Technically, cards in the Black Market deck aren't in the Kingdom.  You aren't considering Prize cards in this, I'm assuming - it's the same principle there.

[Tables]

Unfortunately, the document isn't working for me, but 1 septillion seems out, unless you included every possible Black Market, which while technically correct, is a bit... blah. If you do, then it's probably right, but an incredibly vast majority are virtually identical games. If you don't, well... I'll do some quick calculations here (all done in head, yes I know I'm sitting by a computer I don't care) (pre-post edit: I didn't see the second and third post while making this, but I think asides from the Black Market considerations nothing really changes?)

Colony/Platinum is simply a factor of 2. Yes or No. So that can be factored in later. Young Witch is in 10/157th's of games, correct? And it has about... uh, 45 cards to choose, give or take? Well, on average about 1/3rd of those are already in game, so about 30. So we need to add on 10*30/157 lots of the games we have, then multiply by 2 for colony/platinum, making about 7 quadrillion games?

Black Market really does add a lot, though. If it's there, then there's 147 cards not in the game which may or may not be in the BM deck, which makes roughly 2^147 ~=10^45ish choices (approximating as 2^10=10^3)? Which actually makes me wonder exactly where your number came from, and I wish the document actually worked ¬_¬.

[WanderingWinder]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10. Also, on Isotropic, the Black Market deck is 30 randomly chosen cards, not all the extra cards.

Also, I am not counting differences in Black Market decks as different cases. The exercise is to determine how many possible KINGDOMS exist. The Kingdom and the Black Market deck are independent, except when Young Witch is in the Black Market deck, which is a case I study.

On isotropic, Black market deck has 25 cards, not 30. Which is why I was so confused as to where 30 comes from. I think you should specify how many kingdoms are possible on isotropic, too, because that's not the number of kingdoms in dominion. Your comment on the ordering may be right. But I think the biggest thing is, different Black Market decks makes a big difference, so for any reason that you'd actually care about the number, you want those split out.

[werothegreat]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10. Also, on Isotropic, the Black Market deck is 30 randomly chosen cards, not all the extra cards.

Also, I am not counting differences in Black Market decks as different cases. The exercise is to determine how many possible KINGDOMS exist. The Kingdom and the Black Market deck are independent, except when Young Witch is in the Black Market deck, which is a case I study.

On isotropic, Black market deck has 25 cards, not 30. Which is why I was so confused as to where 30 comes from. I think you should specify how many kingdoms are possible on isotropic, too, because that's not the number of kingdoms in dominion. Your comment on the ordering may be right. But I think the biggest thing is, different Black Market decks makes a big difference, so for any reason that you'd actually care about the number, you want those split out.

Why does iso only use 25 cards for the BM deck?  Aren't you supposed to use every possible card?

[WanderingWinder]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10. Also, on Isotropic, the Black Market deck is 30 randomly chosen cards, not all the extra cards.

Also, I am not counting differences in Black Market decks as different cases. The exercise is to determine how many possible KINGDOMS exist. The Kingdom and the Black Market deck are independent, except when Young Witch is in the Black Market deck, which is a case I study.

On isotropic, Black market deck has 25 cards, not 30. Which is why I was so confused as to where 30 comes from. I think you should specify how many kingdoms are possible on isotropic, too, because that's not the number of kingdoms in dominion. Your comment on the ordering may be right. But I think the biggest thing is, different Black Market decks makes a big difference, so for any reason that you'd actually care about the number, you want those split out.

Why does iso only use 25 cards for the BM deck?  Aren't you supposed to use every possible card?

No. You're supposed to do whatever the heck you want.

[O]

Personally I just place whatever is left from the following set into the BM:

Sea Hag, Ambassador, King's Court, Chapel, Fishing Village, Masquerade, Goons, Mountebank, Ghost Ship

[Davio]

Just add 1 fan card and increase it to a gazillion!

[qmech]

It doesn't matter so much what model we're using for the Black Market, as all we care about is whether the Young Witch is in there.

I can read the chat in the original attachment in LibreOffice, but not any of the calculations.  However, I get a different answer:

Requiring a Prosperity card for a Colony game:

Unmarked Bane: 9 425 875 509 228 462 (9.4... x 10^15)
Marked Bane: 19 688 979 465 082 140 (1.9... x 10^16)

Not requiring a Prosperity card for a Colony game:

Unmarked Bane: 10 254 691 226 062 866 (1.0... x 10^16)
Marked Bane: 21 614 757 429 266 540 (2.1... x 10^16)

As expected, these numbers aren't so much more than (157 choose 10) = 1 871 392 332 785 690 (1.8.. x 10^15).  Accounting for Colonies and a marked Bane can't possibly increase the number of Kingdoms by more than 40 (as removing the Young Witch and Colonies and unmarking the Bane produces an unadorned Kingdom).

[Obi Wan Bonogi]

Personally I just place whatever is left from the following set into the BM:

Sea Hag, Ambassador, King's Court, Chapel, Fishing Village, Masquerade, Goons, Mountebank, Ghost Ship

Oh man, I like your style here.  I love BM and this one is juicy.


On another note: we could nearly double the possible kingdoms if colony/plat was a toggle on/off as opposed to being linked to prosperity...just an idea =D

[werothegreat]

It doesn't matter so much what model we're using for the Black Market, as all we care about is whether the Young Witch is in there.

I can read the chat in the original attachment in LibreOffice, but not any of the calculations.  However, I get a different answer:

Requiring a Prosperity card for a Colony game:

Unmarked Bane: 9 425 875 509 228 462 (9.4... x 10^15)
Marked Bane: 19 688 979 465 082 140 (1.9... x 10^16)

Not requiring a Prosperity card for a Colony game:

Unmarked Bane: 10 254 691 226 062 866 (1.0... x 10^16)
Marked Bane: 21 614 757 429 266 540 (2.1... x 10^16)

As expected, these numbers aren't so much more than (157 choose 10) = 1 871 392 332 785 690 (1.8.. x 10^15).  Accounting for Colonies and a marked Bane can't possibly increase the number of Kingdoms by more than 40 (as removing the Young Witch and Colonies and unmarking the Bane produces an unadorned Kingdom).

What do you mean, a "marked" Bane?

[qmech]

When you're actually playing the game, the Bane is marked.  If you're only interested in the set of 11 cards that are used then I referred to that as an unmarked Bane.  I'm not sure which is a better definition, so I did both.

[qmech]

On another note: we could nearly double the possible kingdoms if colony/plat was a toggle on/off as opposed to being linked to prosperity...just an idea =D

Perhaps surprisingly, it's not that useful.  You have to be fairly unlucky not to draw any Prosperity cards at all in a random Kingdom: it's about a 17.6% chance, which translates to only about an 8% boost in the total number of Kingdoms.

[Davio]

I don't think exact numbers matter here, it's more an Order of Magnitude thing.

Anything above 1 Milliard (1e9) becomes incomprehensible really fast.

So even if it's in the range of 1e15 - making a conservative guess - this still means that it will take 380 million years(!!!) to play every single setup if you play on iso with 5 mins per game (which is pretty quick).

We often talk about combos on these forums and the chance for any 2 cards to be seen together is already pretty slim and gets smaller as more expansions come out. So while these combos are funny little tricks, they don't add that much to the entire strategy domain and may even cause automative thinking.

I just have to tip my hat to the designer that after a year or so (how long has it been?) there are still a lot of kingdoms which can surprise me and leave me with no clue what to do.

[werothegreat]

Anything above 1 Milliard (1e9) becomes incomprehensible really fast.

No one uses long form.  Go back to the dark ages.

[Kirian]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10.

That probability doesn't matter for the number of total kingdoms.  It either has Colony/Platinum, or it doesn't.  Two different kingdoms.

[eHalcyon]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10.

That probability doesn't matter for the number of total kingdoms.  It either has Colony/Platinum, or it doesn't.  Two different kingdoms.

Well, using isotropic rules would prevent kingdoms from having colony/plat if there are no prosperity cards in play.

[kilgoretrout103]

I should have stated: I am assuming Isotropic rules, which means the probability of using Colonies is equal to the number of Prosperity cards in the Kingdom divided by 10.

That probability doesn't matter for the number of total kingdoms.  It either has Colony/Platinum, or it doesn't.  Two different kingdoms.

I agree that the probability doesn't matter. Its only impact is that in a game with ZERO cards from Prosperity, Colonies/Platinums are impossible. Similarly, a game with ONLY Prosperity cards requires Colonies/Platinums. So that reduces the overall number of Kingdoms. (That is, the Kingdom with 10 Prosperity cards but no Colonies doesn't exist.)

EDIT: eHalcyon beat me to it. What he said.

[kilgoretrout103]

It doesn't matter so much what model we're using for the Black Market, as all we care about is whether the Young Witch is in there.

I can read the chat in the original attachment in LibreOffice, but not any of the calculations.  However, I get a different answer:

Requiring a Prosperity card for a Colony game:

Unmarked Bane: 9 425 875 509 228 462 (9.4... x 10^15)
Marked Bane: 19 688 979 465 082 140 (1.9... x 10^16)

Not requiring a Prosperity card for a Colony game:

Unmarked Bane: 10 254 691 226 062 866 (1.0... x 10^16)
Marked Bane: 21 614 757 429 266 540 (2.1... x 10^16)

As expected, these numbers aren't so much more than (157 choose 10) = 1 871 392 332 785 690 (1.8.. x 10^15).  Accounting for Colonies and a marked Bane can't possibly increase the number of Kingdoms by more than 40 (as removing the Young Witch and Colonies and unmarking the Bane produces an unadorned Kingdom).

I'm interested where your numbers come from. I wish you could see my calculations, as I'd like to have an opinion on whether they make sense or not. PM with your email and I can send a PDF.

[ycz6]

Well, using isotropic rules would prevent kingdoms from having colony/plat if there are no prosperity cards in play.

False!

[eHalcyon]

Well, using isotropic rules would prevent kingdoms from having colony/plat if there are no prosperity cards in play.

False!

I stand corrected.

That's from actually specifying Colony/Plat, right?

[Davio]

Anything above 1 Milliard (1e9) becomes incomprehensible really fast.

No one uses long form.  Go back to the dark ages.

Continental Europe does. And it makes more sense than short form.
At least there's no possible confusion with it.

Tell me, how much is a Trillion? (2 answers possible).

[qmech]

I'm interested where your numbers come from. I wish you could see my calculations, as I'd like to have an opinion on whether they make sense or not. PM with your email and I can send a PDF.

I'll do the number of games without Colonies as an example.

Number of Kingdoms without Young Witch: (156 choose 10)
Number of Kingdoms with Young Witch (marked Bane): 42(155 choose 9)
Number of Kingdoms with Young Witch (unmarked Bane): (156 choose 10) - (114 choose 10)
Number of Kingdoms with Black Market -> Young Witch (marked Bane): 41(154 choose 9)
Number of Kingdoms with Black Market -> Young Witch (unmarked Bane): (155 choose 10) - (114 choose 10)

To get the number of Colony games I did the same calculations for all the non-Prosperity cards, then subtracted off the totals.  What I didn't do is subtract off the Prosperity-only games from the non-Colony total, which I should have under the rules you specified.

I can take a look at your calculations if you like.  PM sent.