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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.

[werothegreat]

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).

I realize that "billion" is supposed to mean "million million" and "trillion" is supposed to mean "million million million" and that's where "bi" and "tri" come from, and I realize that it would make more sense if instead of "million, billion, trillion" we had "bousand, trousand, quousand" or something like that, but the half things, the "-ards", while sounding cool, just make the whole process more confusing.  If someone says "quadrilliard," that takes a moment of thought.  If someone says "quadrillion" in short form, you just think "oh, that's the 4th one after thousand".  It might not make sense semantically, but short form is easier to work with mathematically.  And "quadrillion" (short form) is a much more useful number than "quadrillion" (long form).  The former may actually see the light of day with the US's national debt soon, but the latter will probably only ever be used to quantify the size of stars, and at the point, you're going to use scientific notation anyway.

[Kirian]

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).

How is there no possible confusion?  Only someone who uses both systems could possibly confuse "trillion" with anything other than 10^12.

Looking up information on this, interestingly the plurality of the world's population uses neither system; China, India, and Japan, among others, use other traditional systems.  Among the 25 most populous nations, 11 use the short system (billion = 10^9), 9 use a traditional system, and only 5 use the long system.

In other words, except for most of continental Europe, and a few former French and Spanish dependencies (Mexico and Congo, for instance), no one uses long form.  It is approximately as backwards as the USA still using the Imperial measurement system.

[DStu]

In other words, except for most of continental Europe, and a few former French and Spanish dependencies (Mexico and Congo, for instance), no one uses long form.  It is approximately as backwards as the USA still using the Imperial measurement system.

Seems like there opens a path to a compromise...

[Kirian]

In other words, except for most of continental Europe, and a few former French and Spanish dependencies (Mexico and Congo, for instance), no one uses long form.  It is approximately as backwards as the USA still using the Imperial measurement system.

Seems like there opens a path to a compromise...

Oh please, please, please.  I hate the Imperial system.  Please let me use SI units in everyday life.  Please.

[theory]

The metric system is the tool of the devil! My car gets 40 rods to the hogshead and that's the way I likes it.

[werothegreat]

The metric system is the tool of the devil! My car gets 40 rods to the hogshead and that's the way I likes it.

The one big thing that bothers me about the Imperial system is that "pounds" are both a measure of mass, and a measure of force, which is one of the big reasons why most people still have trouble distinguishing the two.  "Hurr hurr, I weight less on the Moon, so I won't have to diet there!"  "... yes, you will, you fut fack."


(wasn't sure if I'm allowed to say naughty words on this forum - and yet spellcheck tells me "fut" is actually a word...)

[WanderingWinder]

The metric system is the tool of the devil! My car gets 40 rods to the hogshead and that's the way I likes it.

The one big thing that bothers me about the Imperial system is that "pounds" are both a measure of mass, and a measure of force, which is one of the big reasons why most people still have trouble distinguishing the two.  "Hurr hurr, I weight less on the Moon, so I won't have to diet there!"  "... yes, you will, you fut fack."


(wasn't sure if I'm allowed to say naughty words on this forum - and yet spellcheck tells me "fut" is actually a word...)

Actually, pounds aren't a unit of mass. You have the lazy 'poundmass', which is the amount of mass to give you a pound of force under 'normal earth gravity', and you have, more properly but less used, slugs.
Also, I'm not sure why you'd think 'naughty words' are unacceptable, but such obvious spoonerisms (is this technically a spoonerism) are fine...

[Schneau]

(wasn't sure if I'm allowed to say naughty words on this forum - and yet spellcheck tells me "fut" is actually a word...)

That's 12 ichnes.

[DStu]

slugs.

Just realized that this imperial system really misses some strange unit for "time"...

[WanderingWinder]

slugs.

Just realized that this imperial system really misses some strange unit for "time"...

What do you mean? Seconds is the strange unit, and originates pre-metric.

[Tables]

On the subject of strange units of time, it looks like WW made that post about 185 ounces ago...

[Davio]

Don't even start about time.

As a software programmer I find myself battling time and awkward APIs every day.

I'm a fan of New Earth Time as it's very international.
230 degrees in The Netherlands is 230 degrees in the USA. It's basically the location of the 12 AM line.
So if the time is 230 degrees, it's 12 AM at 230 degrees latitude (or was it longitude?)

[theory]

I think with all the SI fetishization we miss a couple points: namely, that convenience should trump mathematics for everyday use.

This is an argument in favor of using metric to measure distance (though in construction, imperial tends to be easier because of how many factors 12 and 36 have), but not time, and not temperature.  For everyday use, Fahrenheit is objectively more convenient than Celsius.

[WanderingWinder]

For everyday use, Fahrenheit is objectively more convenient than Celsius.

It is?

[timchen]

For everyday use, Fahrenheit is objectively more convenient than Celsius.

American prejudice! I think it really depends on what you grow up using.

[theory]

Yeah.  Celsius degrees are twice as "fat" as Fahrenheit degrees and therefore less precise with fewer significant digits.

Americans can say "today is in the 60s" and you immediately have a sense of what the temperature is.  You cannot say "today is in the 20s" and know what to expect from a Celsius perspective.

In other words, Fahrenheit is sufficiently precise that you need only one significant digit for everyday purposes.  Celsius requires two.

Plus, 0 to 100 is a convenient frame of reference for the most extreme weather temperatures the average person will likely encounter.

It's true that 0 = freezing and 100 = boiling is somewhat convenient for Celsius purposes.  But in every day life I have never, ever needed to know that 32 was freezing or 212 was boiling.  It just isn't relevant for figuring out today's weather.

[DStu]

I think with all the SI fetishization we miss a couple points: namely, that convenience should trump mathematics for everyday use.

This is an argument in favor of using metric to measure distance (though in construction, imperial tends to be easier because of how many factors 12 and 36 have), but not time, and not temperature.  For everyday use, Fahrenheit is objectively more convenient than Celsius.

Time I also don't see. "Day" ans "Year" are the only one that are fixed, "Month" is quite hard to define /10, but everything else below a day, you could just define as you like.

[theory]

But we're used to 24 hour days and 60 minutes.  It's not especially more convenient to force people to change to decidays and centidays.  And 10 is wonderful for multiplying but not so wonderful for being divided.

[Fabian]

You know what I'm used to though? Celsius. I absolutely don't buy that Fahrenheit is more convenient (never mind "objectively" so, heh), and I think if you were used to using celsius the post above wouldn't have been written.

[timchen]

Yeah.  Celsius degrees are twice as "fat" as Fahrenheit degrees and therefore less precise with fewer significant digits.

Americans can say "today is in the 60s" and you immediately have a sense of what the temperature is.  You cannot say "today is in the 20s" and know what to expect from a Celsius perspective.

In other words, Fahrenheit is sufficiently precise that you need only one significant digit for everyday purposes.  Celsius requires two.

The first two points are true but somewhat contradictory. In fact, both systems are over-precise for weather proposes. Your point is more valid if we compare Fahrenheit/10 with Celcius/10.

And actually, 10s(5-15) in Celsius= cold, 20s= comfortable, 30s= hot also work fine. Or if you figure this is not precise enough you can think in increments of 5. Sure writing down 2 digits is twice as hard as writing one; nevertheless conceptually increment of 5 and 10 are not that different. As I said, it really is a habitual thing.

On the other hand, can anyone tell me why or whether base ten numbers is better than other bases, say, base 8 or base 16?

[Thisisnotasmile]

On the other hand, can anyone tell me why or whether base ten numbers is better than other bases, say, base 8 or base 16?

Because we have ten digits.

[WanderingWinder]

I'm holding out for a factorial-base number system, and (more seriously) an 'absolute' temperature scale.

Indeed, the 1 sig fig argument does not sway me that much. ; saying that the temperature is 70 is a pretty significant difference to me than 79. Not to mention that 1 sig fig in the 0-9 range of the celsius scale is pretty darn useful, significantly moreso than you get from the fahrenheit scale. Finally, and most importantly, the extra effort to go from one digit to two is really negligible - in any case, I want more. So I think the reason that '70s' and the like get used more is that the temperature varies from moment to moment. I get upper/lower/mid much more than just 70s. But anyway, for these purposes, you can do a similar thing on celsius - it's around 23, or it's in the mid 20s, or whatever it is.
I use fahrenheit, because I'm used to it, but I don't think that it's really easier. I just happen to be used to it. Somehow, feet are not more normal to me than metres, though; neither for volume; for weight I know pounds much more - nobody uses newtons day-to-day - but between pound-mass and kilogram... I guess pound-mass might be slightly more normal to me. But not super much. Certainly I'm more used to grams than ounce-equivalent.

[Davio]

It's all about what you learn when you're young.

If we start teaching our kids that we use 20 digits instead of 10, they'll be able to use it flawlessly.
Then again if we have 20 digits, we can't really express the number of digits in base 10.

Neither Fahrenheit nor Celsius (nor Kelvin) is optimal for every day use for simpletons. We could just express the weather temperature on a scale from 0-10 with 0 meaning "freezing your ass off" and 10 meaning "noon in the Sahara desert". This would map from -10 to 40 Celsius or something like that.
In human understanding, there isn't a big difference between -10 and -50, only "very cold" to "colder than the devil's heart". Anything above 40 also becomes incomprehensible, it's only at 100 that we can say, well that's when water boils.

And that's not even true, water at the tip of Mount Everest boils at a 69 degrees Celsius.

[theory]

On the other hand, can anyone tell me why or whether base ten numbers is better than other bases, say, base 8 or base 16?

This was a very helpful post on AskScience Reddit on this question.  Turns out the mathematically "optimal" base is ... e.

[eHalcyon]

And actually, 10s(5-15) in Celsius= cold, 20s= comfortable, 30s= hot also work fine.

In human understanding, there isn't a big difference between -10 and -50, only "very cold" to "colder than the devil's heart".

Whoa, what?  No Canadians here, I guess?  5-15 C is not cold, it's cool.  It's not cold until -20.

And let me tell you, there is a huge difference between -10 and -40 (not going to consider -50, haven't experienced that).  In -10 weather I'm still wearing t-shirts under my fall jacket, maybe with a hoodie.  At -40 I'm wearing long johns and have a giant winter jacket.  -10 is still pleasant.  -40 is not.

[WanderingWinder]

And actually, 10s(5-15) in Celsius= cold, 20s= comfortable, 30s= hot also work fine.

In human understanding, there isn't a big difference between -10 and -50, only "very cold" to "colder than the devil's heart".

Whoa, what?  No Canadians here, I guess?  5-15 C is not cold, it's cool.  It's not cold until -20.

And let me tell you, there is a huge difference between -10 and -40 (not going to consider -50, haven't experienced that).  In -10 weather I'm still wearing t-shirts under my fall jacket, maybe with a hoodie.  At -40 I'm wearing long johns and have a giant winter jacket.  -10 is still pleasant.  -40 is not.

Conversely I find 25 C cool. Even 10 C is cold to me. And it's not hot until you hit 35, in all seriousness.  I was stunned that the ESPN guys doing France/England called 86 F "sweltering"...

[eHalcyon]

Conversely I find 25 C cool. Even 10 C is cold to me. And it's not hot until you hit 35, in all seriousness.  I was stunned that the ESPN guys doing France/England called 86 F "sweltering"...

You got me there.  25 C is getting rather warm for me.  I would totally consider 86 F (30 C) sweltering. 

[kilgoretrout103]

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.

Ah, crap, you're right. I missed the "marked Bane" distinction, so I treated "10 cards, where Hamlet is in the Kingdom normally and Watchtower is the Bane" as a different case from "10 cards, where Watchtower is in the Kingdom normally and Hamlet is the Bane." So that would significantly reduce the number of possibilities.

[blueblimp]

On the other hand, can anyone tell me why or whether base ten numbers is better than other bases, say, base 8 or base 16?

This was a very helpful post on AskScience Reddit on this question.  Turns out the mathematically "optimal" base is ... e.

That's for a certain model of how computers work. For humans, I figure you want the base to be as big as it can be while still having a reasonably-sized times table, and then it's nice to have convenient factors as a bonus. Since people often learn up to 12x12 anyway, we should be safe up to base 13, but 12 has nicer factors than 13.

[Ozle]

I cant believe there is no exact soecific answer posted yet! Or did i miss it?

[kilgoretrout103]

I cant believe there is no exact soecific answer posted yet! Or did i miss it?

Assuming we're willing to count the same set of 11 cards--but with different cards as the Bane--as different Kingdoms, then I think my original answer of 1.076 x 10^24 holds. Of course, that assumes a 30 card BM deck and strict Prosperity/Colony rules. Modifying those conditions will obviously give you different answers.

[qmech]

I cant believe there is no exact soecific answer posted yet! Or did i miss it?

With a free choice of whether or not to use Colonies and with a distinguished Bane there are 21 614 757 429 266 540 = 2.1... x 10^16 distinct Kingdoms.  (The "Isotropic rules" counts are off because I included Prosperity only games in the non-Colony count.)

It can't be anywhere near 10^24 because 80x(157 choose 10) is a generous upper bound.

[Edit: added extra factor of 2 to cover the cases where Young Witch is in the Black Market deck.]

Back 'on' topic: 15C is warm.  At 25C the good people of Britain collapse in puddles of sweat and fail to get any work done (which is handy because the trains have probably also broken down).

[GendoIkari]

I cant believe there is no exact soecific answer posted yet! Or did i miss it?

Assuming we're willing to count the same set of 11 cards--but with different cards as the Bane--as different Kingdoms, then I think my original answer of 1.076 x 10^24 holds. Of course, that assumes a 30 card BM deck and strict Prosperity/Colony rules. Modifying those conditions will obviously give you different answers.

I definitely think that you should count that same set of 11 cards as a different Kingdom if the Bane is different. Mostly because the strategies could play out very different between the 2.

[werothegreat]

The metric system is the tool of the devil! My car gets 40 rods to the hogshead and that's the way I likes it.

The one big thing that bothers me about the Imperial system is that "pounds" are both a measure of mass, and a measure of force, which is one of the big reasons why most people still have trouble distinguishing the two.  "Hurr hurr, I weight less on the Moon, so I won't have to diet there!"  "... yes, you will, you fut fack."


(wasn't sure if I'm allowed to say naughty words on this forum - and yet spellcheck tells me "fut" is actually a word...)

Actually, pounds aren't a unit of mass. You have the lazy 'poundmass', which is the amount of mass to give you a pound of force under 'normal earth gravity', and you have, more properly but less used, slugs.
Also, I'm not sure why you'd think 'naughty words' are unacceptable, but such obvious spoonerisms (is this technically a spoonerism) are fine...

Apparently naughty words aren't acceptable on here, because my naughty post was deleted.  At any rate, I love how derailed this thread has gotten.

[kilgoretrout103]

I cant believe there is no exact soecific answer posted yet! Or did i miss it?

Final answer! A giant thank you to qmech, who pointed out two things:

 (1) The probability of YW in the BM is irrelevant, and they should both be weighted as 1. (This is a minor change to my original formula)

(2) More importantly, I made a typo when I calculated the results of my formula! It should not be so large. So, without further ado, here is the final answer:

1.2274 x 10^16, or about 12 quadrillion. Much more in line with the upper-bound posited by qmech. Woo!

EDIT: Here is my formula, for those who care:

Case 1: No BM, No YW, No Colonies = (155 choose 10) - (25 choose 10)
Case 2: BM, No YW, No Colonies = 41(154 choose 9)
Case 3: BM, No YW, No Colonies = (155 choose 9) + 41(154 choose 9)
Case 4: No BM, No YW, Colonies = (155 choose 10) - (130 choose 10)
Case 5: BM, YW, No Colonies = 41(154,8)
Case 6: BM, YW, Colonies = 3(154 choose 8 ) + 38*((154 choose 8 ) - (130 choose 8 ))

case1+case2+case3+case4+case5+case6 = 12274208207648045 = About 12 quadrillion

EDIT #2: Getting those formulas to appear without emoticons was non-trivial, haha.

[theory]

And what are the assumptions made in this 12 quadrillion number?

[WanderingWinder]

I cant believe there is no exact soecific answer posted yet! Or did i miss it?

With a free choice of whether or not to use Colonies and with a distinguished Bane there are 21 614 757 429 266 540 = 2.1... x 10^16 distinct Kingdoms.  (The "Isotropic rules" counts are off because I included Prosperity only games in the non-Colony count.)

It can't be anywhere near 10^24 because 80x(157 choose 10) is a generous upper bound.

[Edit: added extra factor of 2 to cover the cases where Young Witch is in the Black Market deck.]

Back 'on' topic: 15C is warm.  At 25C the good people of Britain collapse in puddles of sweat and fail to get any work done (which is handy because the trains have probably also broken down).

Nonsense. 25C is the exact temperature, to degree celsius, I would keep the interior of my house at, were heating/cooling costs not an issue.

[kilgoretrout103]

And what are the assumptions made in this 12 quadrillion number?

Only that the probability of playing with Colonies is equal to the number of Prosperity cards in the Kingdom (including the Bane) divided by 10.

It works for any size Black Market deck (provided it could possibly contain Young Witch).

I count the same Kingdom with different Black Market decks as the same case, but I count the same 11 cards where a different one of those 11 is the bane as different cases.

EDIT: Upon further reflection, if we follow the Prosperity recommendation to the letter, the Bane shouldn't affect the probability of Colonies. If we go by that instead, we change case 6 to:

Case 6: BM, YW, and Colonies = 41*((154 choose 8 ) - (130 choose 8 )), and there are 4,872,533,562,000 fewer possible kingdoms. So then our answer is 1.2269 x 10^16 instead of 1.2274 x 10^16.

[WanderingWinder]

Only that the probability of playing with Colonies is equal to the number of Prosperity cards in the Kingdom (including the Bane) divided by 10. nonzero IFF there is at least one prosperity card, and 1 if all kingdom cards are from prosperity.

It works for any size Black Market deck (provided it could possibly contain Young Witch).

I count the same Kingdom with different Black Market decks as the same case, but I count the same 11 cards where a different one of those 11 is the bane as different cases.

The exact probabilities don't matter, except to break down into 3 cases: 0, 1, or between.
I also don't think you're accounting for what happens if all 2s/3s are in black market along with young witch existing either, though apparently it's not clear what happens in this case....

[kilgoretrout103]

The exact probabilities don't matter, except to break down into 3 cases: 0, 1, or between.
I also don't think you're accounting for what happens if all 2s/3s are in black market along with young witch existing either, though apparently it's not clear what happens in this case....

Interesting thought.

Under my original assumption of a size 30 (or 25) Black Market deck, this is impossible because there are fewer cards in the Black Market deck than potential Banes. But let's say that assumption doesn't hold, and the BM deck consists of every card not in the Kingdom. Then, the only ambiguous case occurs when Black Market is in the Kingdom, but Young Witch isn't. You pick your 10 Kingdom cards, throw the rest in the Black Market deck, go to add the Bane because YW is in the BM deck, but oh look, you can't, they're all in the BM deck.

I don't know of any official rules that cover this case, but if I were making the call, I'd say choose a Bane at random from the BM deck, and put it in the Kingdom instead. If that's our solution, it shouldn't change our count.

[Piemaster]

If we assume that you just chuck every non-kingdom card into the BM (or simply that the choice of what to put in the BM falls outside the scope of the calculation seeing as it is non-random) then this discussion becomes a hell of a lot easier and we might have a prayer at finding an answer.

[Davio]

Don't forget that BM is a Promo card, we keep treating it like it's an official card from one of the expansions.

You shouldn't take Promo cards too seriously.

If you're ever in the situation that all $2 and $3 cards are in the BM, just take a random one out and put a random higher cost in.

Not everything needs a ruling.

[Piemaster]

Don't forget that BM is a Promo card, we keep treating it like it's an official card from one of the expansions.

You shouldn't take Promo cards too seriously.

Really?  I have never seen anything to suggest that promo cards are anything other than legit and balanced cards (albeit thematic ones)

If there was a World Championships do you think they would be included?

[Tables]

I believe the World Championships used the base game only .

Anyway, what's this nonsense about 25 degrees being a sensible temperature? In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot. And it's taken the temperature being in single digits before I've decided I need to put something extra back on. 25 degrees, man, that IS swelteringly hot.

Whether Black Market is a serious card or not, it's certainly an option for being in the game, and leads itself to two possibilities: BM contains YW, BM doesn't contain YW (whose probabilities don't matter as long as they aren't 0 or 1 - and it's only 0 if the kingdom contains YW). I think (at a glance) 1.2*10^16 looks right. However, it's of course possible to play with Colony/Platinum without Prosperity. Donald has I believe said it's okay and the recommendation is only that, so I believe such games should be counted.

[WanderingWinder]

Anyway, what's this nonsense about 25 degrees being a sensible temperature? In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot. And it's taken the temperature being in single digits before I've decided I need to put something extra back on. 25 degrees, man, that IS swelteringly hot.

You're insane . Remind me to bring a jacket if I go to your house.
25 is totally sensible. I've lived entire summers at 27 without problems really. My totally honestly ideal temperature is between 24 and 25, but probably closer to the 25 side. At 18, I usually need to put more clothes on.

[eHalcyon]

Anyway, what's this nonsense about 25 degrees being a sensible temperature? In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot. And it's taken the temperature being in single digits before I've decided I need to put something extra back on. 25 degrees, man, that IS swelteringly hot.

I'm Canadian and this is still too much for me.  I think my ideal room temperature is ~19-22.  Do you live extra far north? 

[clb]

Just so WW doesn't feel all alone - I grew up in southern Arizona, and 25 is a very normal temperature for the thermostat. The electric company would encourage us to set it lower in the winter to conserve energy, but that is a seperate issue.
As to outdoor temperatures, 20-30 is very comfortable, in fact, 30 is as cool as it gets overnight for 5+ months of the year. 35-37 is warm, but we often have 100+ days in a year of highs over 37. 40-43 is where people start complaining about the heat.
On the flipside, 10 was considered pretty cool weather and time to don a sweater (or a jacket if you will be outdoors for long). Now that I live in the Seattle area (almost to the border with our Canadian friends), when the temperature hits 10 in April or so, its time for shorts, t-shirt and maybe some flip-flops/sandals.

[Ozle]

Can only really go by inside (home/Office) temps

21 seems reasonable to me, thats what i would set my car or air conditioning at for comfort.
But anywhere between 18 and 24 is also acceptable to me.
25 I would describe as a bit Warm in the office
And at 27 I would probably be moaning about the heat

[Kuildeous]

Anyway, what's this nonsense about 25 degrees being a sensible temperature? In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot.

That must have made for an awkward holiday dinner.

[Kirian]

Yeah, 25 is doable but a bit higher than I would like it.  We generally keep the thermostat near 23 in summer and 20 in winter.  Anything above 27 means I stay in the AC or a pool.  Anything below 20 and I put on a light jacket.

I guess that's what comes of growing up at the border of a Dfa and a Cfa climate (NW Ohio). 

[WanderingWinder]

Yeah, 25 is doable but a bit higher than I would like it.  We generally keep the thermostat near 23 in summer and 20 in winter.  Anything above 27 means I stay in the AC or a pool.  Anything below 20 and I put on a light jacket.

I guess that's what comes of growing up at the border of a Dfa and a Cfa climate (NW Ohio). 

From what I can tell, this is pretty much average.

[qmech]

 

In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot.

It's nice to have sane company.  To throw some more fuel on the fire, I grew up in a house with the thermostat set to 13C.

[Tables]

Anyway, what's this nonsense about 25 degrees being a sensible temperature? In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot. And it's taken the temperature being in single digits before I've decided I need to put something extra back on. 25 degrees, man, that IS swelteringly hot.

I'm Canadian and this is still too much for me.  I think my ideal room temperature is ~19-22.  Do you live extra far north? 

Nope, in the south of England (for most sensible definitions of 'south' - my housemate considers e.g. London pretty far north) although I'll admit I probably exaggerated a little. It might have been closer to 20 (the thermometer's pretty inaccurate) and when I say taking clothes off, I did just mean a hoodie, nothing innapropriate (get your mind out of the gutter, Kuleidos!). Still, 25 degrees is too much, 15 is just right.

[Morgrim7]

You guys are crazy. 20 is perfectly fine, 25 is hot, 30 is horrid. 40 does call for some awkwad meals, Kuildeous.

[eHalcyon]

Anyway, what's this nonsense about 25 degrees being a sensible temperature? In my house, we stay at about 15 degrees. I've walked in once with it being 18 degrees and immediately taken clothes off for it being too hot. And it's taken the temperature being in single digits before I've decided I need to put something extra back on. 25 degrees, man, that IS swelteringly hot.

I'm Canadian and this is still too much for me.  I think my ideal room temperature is ~19-22.  Do you live extra far north? 

Nope, in the south of England (for most sensible definitions of 'south' - my housemate considers e.g. London pretty far north) although I'll admit I probably exaggerated a little. It might have been closer to 20 (the thermometer's pretty inaccurate) and when I say taking clothes off, I did just mean a hoodie, nothing innapropriate (get your mind out of the gutter, Kuleidos!). Still, 25 degrees is too much, 15 is just right.

Oh, if you're wearing a hoodie, that changes things.  It's not hoodie weather until below 0.

[cayvie]

It was weird moving from North Carolina to Minnesota and feeling my body acclimate itself to the temperature shift over time.

I used to be able to tell 80 F from 85 F just by walking outside, and anything below 30 was just "cold".

Now I can tell 10 F from 15 F no problem, but anything above 80 is just "hot".