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Author Topic: The Dominion Number  (Read 15857 times)

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Asubfive

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The Dominion Number
« on: September 16, 2012, 10:28:53 am »
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We often ask challenges like this: Maximize/minimize n such that there is a kingdom k such that P(k,n).

This is a question of the form: Maximize/minimize n such that for all kingdoms k we have P(k,n).

As remarked by Donald, one of the most obvious choice for P is "A solo player in kingdom k can gain all Provinces in n turns."

Therefore, I ask: What is the minimal n such that a solo player can guarantee gaining all Provinces in n turns no matter what the kingdom?

This minimal n could be called "The Dominion Number" akin to God's Number for Rubik's Cube.

Clarifications:
  • The shuffling is random.
  • Extra turns like Outpost and Possession turns do not count (but I doubt it matters for the solution).
  • Any Black Market deck is allowed (but I also doubt it matters).
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DStu

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Re: The Dominion Number
« Reply #1 on: September 16, 2012, 11:01:00 am »
+4

Random shuffling means worst case shuffling, as otherwise you can't guarantee that you don't have worse case shuffle luck. And for all kingdoms means without kingdom cards, as we already have a kingdom where bm is optimal.
So the question is how many turns it takes bm to get all provinces with worst shuffle luck
Edit. The proof is even easier here, as we only want Provinces, you could fill the kingdom with altenate Victories, which obviously don't help.
Gardens, Duke, Great Hall, Harem, Vineyard, Silk Road, Feodum, Fairground will all not help. Add Sea Hag and Saboteur, and you have a Kingdom with no cards that will help your BM to get Provinces
« Last Edit: September 16, 2012, 11:11:00 am by DStu »
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Davio

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Re: The Dominion Number
« Reply #2 on: September 16, 2012, 01:21:58 pm »
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Yeah, random shuffling means we're looking for a big Oh kind of thing.
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blueblimp

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Re: The Dominion Number
« Reply #3 on: September 16, 2012, 01:48:29 pm »
+3

Getting all Provinces with pure BM with worst-case shuffling luck sounds very unpleasant. So many 5's and 7's. :P
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Re: The Dominion Number
« Reply #4 on: September 16, 2012, 02:34:27 pm »
+2

Getting all Provinces with pure BM with worst-case shuffling luck sounds very unpleasant. So many 5's and 7's. :P

Indeed, just determining worst-case shuffling is going to be a PITA.  With worst-case shuffling I would guess we can put off the first Gold by until at least T8 or T9, and delay the first $8 until probably T16 or later.  Let's see:

Open 2/5 for Silver/-
T3: SCCCE, buy Silver
T4: CCCCE, buy Silver
T5: E|SCCE, buy Silver (Deck is 2S, 5C, E)
T6: SCCCE, buy Silver
T7: SCCE|C, buy Silver (Deck is 4S, 4C, 2E)
T8: SCCCE, buy Silver
T9: SSSCE, buy Gold

Can anyone do worse?
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Re: The Dominion Number
« Reply #5 on: September 16, 2012, 04:39:20 pm »
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This probably requires you to get the average worth of cards in your deck over the magical 5/8 = 1.6, since otherwise you can probably go through a shuffle with all $7 hands. Once you are over 1.6, you are guaranteed at least one Province during one time through your deck. Of course, with worst shuffle luck your hand might be 5 Golds, and the rest of your hands being awful. So, it might end up being only one Province per shuffle with worst shuffle luck.
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Kirian

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Re: The Dominion Number
« Reply #6 on: September 16, 2012, 04:47:03 pm »
+1

This probably requires you to get the average worth of cards in your deck over the magical 5/8 = 1.6, since otherwise you can probably go through a shuffle with all $7 hands. Once you are over 1.6, you are guaranteed at least one Province during one time through your deck. Of course, with worst shuffle luck your hand might be 5 Golds, and the rest of your hands being awful. So, it might end up being only one Province per shuffle with worst shuffle luck.

In addition, every Province buy decreases the average card strength.  The question becomes:  is it going to be faster to exceed that average, or to hit that average, buy a Province, buy a Gold, etc.?
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eHalcyon

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Re: The Dominion Number
« Reply #7 on: September 16, 2012, 05:00:22 pm »
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This probably requires you to get the average worth of cards in your deck over the magical 5/8 = 1.6, since otherwise you can probably go through a shuffle with all $7 hands. Once you are over 1.6, you are guaranteed at least one Province during one time through your deck. Of course, with worst shuffle luck your hand might be 5 Golds, and the rest of your hands being awful. So, it might end up being only one Province per shuffle with worst shuffle luck.

In addition, every Province buy decreases the average card strength.  The question becomes:  is it going to be faster to exceed that average, or to hit that average, buy a Province, buy a Gold, etc.?

The Province will decrease average value, but all your subpar hands during that shuffle will probably be used to buy more Silver, which should balance out or even push it in your favour when you reshuffle... right?
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Kirian

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Re: The Dominion Number
« Reply #8 on: September 16, 2012, 06:11:01 pm »
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This probably requires you to get the average worth of cards in your deck over the magical 5/8 = 1.6, since otherwise you can probably go through a shuffle with all $7 hands. Once you are over 1.6, you are guaranteed at least one Province during one time through your deck. Of course, with worst shuffle luck your hand might be 5 Golds, and the rest of your hands being awful. So, it might end up being only one Province per shuffle with worst shuffle luck.

In addition, every Province buy decreases the average card strength.  The question becomes:  is it going to be faster to exceed that average, or to hit that average, buy a Province, buy a Gold, etc.?

The Province will decrease average value, but all your subpar hands during that shuffle will probably be used to buy more Silver, which should balance out or even push it in your favour when you reshuffle... right?

Actually, you would only want to buy just enough Silver/Gold to bring you back across that threshold, lest you increase the number of turns between shuffles unnecessarily.

I'm thinking that we're looking at N = 40 at least here.
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Tables

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Re: The Dominion Number
« Reply #9 on: September 16, 2012, 07:19:19 pm »
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I've worked this one out before. Well, almost. I'll spoiler anyway even though I answer a slightly different question (worst case time for the standard BMU bot ignoring Duchies to get to 4 Provinces)
If I remember correctly, it's 35 to get to 4 Provinces, assuming we follow the standard procedure of buying a Province iff total coin in deck >=17. Obviously, the idea is we hit 5 as often as possible, wasting as much money as possible. However, once we get 5 Silver we can do even better, massively overkilling on some Provinces and getting three Provinces ASAP.

However, there are two problems here. Firstly, if we're going for worst case time, we WON'T pick up those early Provinces, because we know they'll clog our deck up too much. So we make our deck better instead, which means Golds over Provinces and generally setting a higher bar before we start greening. Secondly, we're aiming for 8 Provinces not 4, but again, that just means we set a higher bar before starting.

Actually doing the maths and working out the exact optimum point to switch from Golds to Provinces will be tricky, as we have to bear in mind all kinds of annoying 'worst case' tricks (like, having to buy Golds for $13 switches into missing Provinces and buying Golds with $7)


And considering this, I think Kirian is right. It's almost certainly at least 40 turns, probably even more than that.
« Last Edit: September 16, 2012, 07:20:56 pm by Tables »
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Powerman

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Re: The Dominion Number
« Reply #10 on: September 16, 2012, 10:00:14 pm »
+1

I think a good starting point would be to run a bot with simple buy rules for a ton of simulations and go from there.
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DG

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Re: The Dominion Number
« Reply #11 on: September 16, 2012, 10:12:27 pm »
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It's reassuring to know that in a four player game, the players cannot get stuck indefinitely with repeated bad draws that cannot muster 8 coins for a province. If the gold and silver piles are exhausted then someone will eventually get a province buying hand.
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jomini

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Re: The Dominion Number
« Reply #12 on: September 16, 2012, 10:23:14 pm »
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It's reassuring to know that in a four player game, the players cannot get stuck indefinitely with repeated bad draws that cannot muster 8 coins for a province. If the gold and silver piles are exhausted then someone will eventually get a province buying hand.

That is true only for simple BM games. Something simple, like adding in minion, can make folks miss their cards and have to make 8 coin in only 4 cards (worst case scenario with militia could be even worse with hands like militia/silver x3/gold cropping up again and again). Likewise swindler may be able to trash all the silver/gold without replacement nor allow any player to hit 8 coin again. Worst case luck with something like apprentice could also deplete the treasures in deck without allowing for game ending.

The bigger thing is that if you deplete two treasure piles you are going to have to have a LOT of 5 hands and the duchy pile will go away long before worst case luck will let the provinces linger (even if the the 3 guys in losing positions refuse to buy duchies). You'd have to really try to force people to get stuck without options for emptying the duchy pile.
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Kirian

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Re: The Dominion Number
« Reply #13 on: September 16, 2012, 11:12:52 pm »
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I think a good starting point would be to run a bot with simple buy rules for a ton of simulations and go from there.

The thing is, a bot won't find the worst-case scenario here.  You have to apply the worst possible shuffle luck in order to delay the first Province buy, but a bot can't be programmed to create "worst possible shuffle luck."  And as long as the average card value is under 7/5, you should be able to get through an entire shuffle without hitting $8.
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greatexpectations

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Re: The Dominion Number
« Reply #14 on: September 16, 2012, 11:32:57 pm »
+1

The thing is, a bot won't find the worst-case scenario here.  You have to apply the worst possible shuffle luck in order to delay the first Province buy, but a bot can't be programmed to create "worst possible shuffle luck."  And as long as the average card value is under 7/5, you should be able to get through an entire shuffle without hitting $8.

i think powerman was suggesting something different. more along the lines of running a ton of simulations, isolating the worst few performances, and working from there.
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Sakako

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Re: The Dominion Number
« Reply #15 on: September 17, 2012, 12:13:08 am »
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You could run a bot for, say, a thousand simulations or so on every possible kingdom, and take the upper bound of solo performances. But of course, who wants to do that? And what about in games which use solely cards from Prosperity? They must, by the rules, be Colony games, so Platinums will also be involved, perhaps making for a shorter game.

Of course, it's true that on many kingdoms, Big Money is the worst strategy, so you could work out the worst-shuffle timing for someone to get a province using solely Big Money for a good approximation.
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Powerman

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Re: The Dominion Number
« Reply #16 on: September 17, 2012, 12:29:52 am »
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The thing is, a bot won't find the worst-case scenario here.  You have to apply the worst possible shuffle luck in order to delay the first Province buy, but a bot can't be programmed to create "worst possible shuffle luck."  And as long as the average card value is under 7/5, you should be able to get through an entire shuffle without hitting $8.

i think powerman was suggesting something different. more along the lines of running a ton of simulations, isolating the worst few performances, and working from there.

Exactly.  Have a bot with buy rules of:
8+: Province
6,7: Gold
3,4,5: Silver

And run (say) 100,000 simulations to see how long it takes for the bot to buy 8 provinces.  Then the worst results can be modified to make it entirely "bad luck".  It will at least give us a starting point to go from.
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blueblimp

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Re: The Dominion Number
« Reply #17 on: September 17, 2012, 01:11:08 am »
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This probably requires you to get the average worth of cards in your deck over the magical 5/8 = 1.6, since otherwise you can probably go through a shuffle with all $7 hands. Once you are over 1.6, you are guaranteed at least one Province during one time through your deck. Of course, with worst shuffle luck your hand might be 5 Golds, and the rest of your hands being awful. So, it might end up being only one Province per shuffle with worst shuffle luck.
I think it's worse, because bad shuffle luck will cause your best treasures to miss the shuffle. So you might need to go through your whole deck about twice per Province, even if your money density is good enough.
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DStu

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Re: The Dominion Number
« Reply #18 on: September 17, 2012, 01:29:12 am »
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The thing is, a bot won't find the worst-case scenario here.  You have to apply the worst possible shuffle luck in order to delay the first Province buy, but a bot can't be programmed to create "worst possible shuffle luck."  And as long as the average card value is under 7/5, you should be able to get through an entire shuffle without hitting $8.

i think powerman was suggesting something different. more along the lines of running a ton of simulations, isolating the worst few performances, and working from there.

Exactly.  Have a bot with buy rules of:
8+: Province
6,7: Gold
3,4,5: Silver

And run (say) 100,000 simulations to see how long it takes for the bot to buy 8 provinces.  Then the worst results can be modified to make it entirely "bad luck".  It will at least give us a starting point to go from.
I don't think we will get even near, but for the lulz:
500.000.000 sims, maximum is 40 turns: record is 40 (3 times)
Number of turns:  Money on turn #
Code: [Select]
40:  4 3 5 6 4 8 5 4 8 5 5 4 10 5 5 5 7 7 3 8 9 5 10 7 6 7 4 12 7 7 5 7 7 7 6 7 5 6 6 14
40:  2 5 5 3 8 4 2 8 5 5 1 8 8 4 5 2 4 9 4 8 4 5 8 2 4 6 7 5 4 6 7 7 6 6 7 5 6 7 6 9
40:  4 3 4 4 4 8 4 5 6 9 5 5 5 4 8 9 5 5 4 11 5 7 5 5 7 7 7 7 6 9 7 6 11 6 7 5 7 6 7 9
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blueblimp

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Re: The Dominion Number
« Reply #19 on: September 17, 2012, 02:53:20 am »
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The thing is, a bot won't find the worst-case scenario here.  You have to apply the worst possible shuffle luck in order to delay the first Province buy, but a bot can't be programmed to create "worst possible shuffle luck."  And as long as the average card value is under 7/5, you should be able to get through an entire shuffle without hitting $8.

i think powerman was suggesting something different. more along the lines of running a ton of simulations, isolating the worst few performances, and working from there.

Exactly.  Have a bot with buy rules of:
8+: Province
6,7: Gold
3,4,5: Silver

And run (say) 100,000 simulations to see how long it takes for the bot to buy 8 provinces.  Then the worst results can be modified to make it entirely "bad luck".  It will at least give us a starting point to go from.
I don't think we will get even near, but for the lulz:
500.000.000 sims, maximum is 40 turns: record is 40 (3 times)
Number of turns:  Money on turn #
Code: [Select]
40:  4 3 5 6 4 8 5 4 8 5 5 4 10 5 5 5 7 7 3 8 9 5 10 7 6 7 4 12 7 7 5 7 7 7 6 7 5 6 6 14
40:  2 5 5 3 8 4 2 8 5 5 1 8 8 4 5 2 4 9 4 8 4 5 8 2 4 6 7 5 4 6 7 7 6 6 7 5 6 7 6 9
40:  4 3 4 4 4 8 4 5 6 9 5 5 5 4 8 9 5 5 4 11 5 7 5 5 7 7 7 7 6 9 7 6 11 6 7 5 7 6 7 9
Interesting that these allow the bot to buy an early Province, introducing a dead card into the deck and wasting money that could have been spent on treasure. I know that BMU teaches that early Provinces are bad, but it's neat to see a random simulation pick up on it.
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DStu

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Re: The Dominion Number
« Reply #20 on: September 17, 2012, 03:30:49 am »
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The thing is, a bot won't find the worst-case scenario here.  You have to apply the worst possible shuffle luck in order to delay the first Province buy, but a bot can't be programmed to create "worst possible shuffle luck."  And as long as the average card value is under 7/5, you should be able to get through an entire shuffle without hitting $8.

i think powerman was suggesting something different. more along the lines of running a ton of simulations, isolating the worst few performances, and working from there.

Exactly.  Have a bot with buy rules of:
8+: Province
6,7: Gold
3,4,5: Silver

And run (say) 100,000 simulations to see how long it takes for the bot to buy 8 provinces.  Then the worst results can be modified to make it entirely "bad luck".  It will at least give us a starting point to go from.
I don't think we will get even near, but for the lulz:
500.000.000 sims, maximum is 40 turns: record is 40 (3 times)
Number of turns:  Money on turn #
Code: [Select]
40:  4 3 5 6 4 8 5 4 8 5 5 4 10 5 5 5 7 7 3 8 9 5 10 7 6 7 4 12 7 7 5 7 7 7 6 7 5 6 6 14
40:  2 5 5 3 8 4 2 8 5 5 1 8 8 4 5 2 4 9 4 8 4 5 8 2 4 6 7 5 4 6 7 7 6 6 7 5 6 7 6 9
40:  4 3 4 4 4 8 4 5 6 9 5 5 5 4 8 9 5 5 4 11 5 7 5 5 7 7 7 7 6 9 7 6 11 6 7 5 7 6 7 9
Interesting that these allow the bot to buy an early Province, introducing a dead card into the deck and wasting money that could have been spent on treasure. I know that BMU teaches that early Provinces are bad, but it's neat to see a random simulation pick up on it.

Waiting until you have 3 Golds gives you (10x50.000.000)
Code: [Select]
37:  3 4 4 4 5 8 4 6 7 10 5 5 9 6 8 5 5 12 10 6 3 5 7 7 12 5 5 14 7 6 7 6 5 7 7 5 9
37:  5 2 5 4 5 6 5 7 6 6 10 9 1 11 12 4 4 7 5 12 5 5 5 7 10 7 7 7 6 6 7 7 7 8 6 7 12
37:  3 4 5 5 5 4 8 5 5 9 8 5 5 4 11 12 6 9 12 4 5 6 10 9 6 5 7 7 7 6 7 7 13 6 7 4 8
38:  4 3 5 5 5 7 5 5 7 10 9 5 5 6 8 10 6 4 12 5 9 7 6 7 7 10 5 6 7 15 7 4 7 7 7 6 3 10
39:  5 2 5 4 2 6 9 5 2 7 10 8 7 5 5 9 12 2 7 5 10 2 10 3 7 7 4 5 7 7 7 7 7 7 5 7 5 9 11
37:  3 4 2 5 6 5 3 7 8 5 8 9 5 8 12 1 4 5 11 7 5 7 7 3 7 13 6 4 6 7 12 7 7 7 7 6 8
38:  3 4 5 4 5 7 4 5 9 7 6 8 8 10 5 4 13 7 5 4 5 10 3 7 6 7 11 5 7 7 7 6 7 14 7 7 3 14
37:  4 3 4 3 5 5 5 5 7 8 10 5 5 5 12 9 10 8 5 8 5 7 6 6 5 11 5 12 5 7 7 7 7 7 7 7 10
37:  3 4 5 5 3 7 5 8 6 10 8 2 9 9 3 5 7 5 5 7 8 7 4 6 7 11 7 7 7 7 6 7 12 7 6 7 10
38:  5 2 4 4 5 4 5 7 5 9 7 5 3 9 8 10 6 9 5 4 10 8 6 6 7 7 7 7 7 7 7 13 6 6 6 5 7 10
3 early Golds followed by 2 early Provinces.  Overall, they all get 6-7 Provinces quite quickly, followed by long sequences of 7s until you get the last.
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RyanRomanik

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Re: The Dominion Number
« Reply #21 on: September 17, 2012, 05:45:01 am »
0


Is there any chance that buying copper at 0-2 would help? I know it's normally bad, but since we're basically assuming all of our good cards are going to be constantly missing reshuffles, maybe it would increase the average $/hand of our effective deck.
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Re: The Dominion Number
« Reply #22 on: September 17, 2012, 07:17:42 am »
0

This probably requires you to get the average worth of cards in your deck over the magical 5/8 = 1.6, since otherwise you can probably go through a shuffle with all $7 hands. Once you are over 1.6, you are guaranteed at least one Province during one time through your deck. Of course, with worst shuffle luck your hand might be 5 Golds, and the rest of your hands being awful. So, it might end up being only one Province per shuffle with worst shuffle luck.
I think it's worse, because bad shuffle luck will cause your best treasures to miss the shuffle. So you might need to go through your whole deck about twice per Province, even if your money density is good enough.

This is true, but maybe the player can combat this by not buying Silvers when unnecessary to try to have a multiple of 5 cards in their deck every reshuffle? Actually, having 0, 1, or 2 cards missing the reshuffle shouldn't be too big of a deal, so the player could try to get those most of the time instead of the bad 3 or 4 cards missing. Since there's no drawing, this is relatively easy to do, but it's unclear how much it will hurt to miss those Silver buys.
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Davio

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Re: The Dominion Number
« Reply #23 on: September 17, 2012, 07:29:21 am »
0

It's a fun challenge: optimizing BM while trying to get poor results. :)
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Re: The Dominion Number
« Reply #24 on: September 17, 2012, 08:24:03 am »
0

Waiting until you have 3 Golds gives you (10x50.000.000)
Code: [Select]
37:  3 4 4 4 5 8 4 6 7 10 5 5 9 6 8 5 5 12 10 6 3 5 7 7 12 5 5 14 7 6 7 6 5 7 7 5 9
37:  5 2 5 4 5 6 5 7 6 6 10 9 1 11 12 4 4 7 5 12 5 5 5 7 10 7 7 7 6 6 7 7 7 8 6 7 12
37:  3 4 5 5 5 4 8 5 5 9 8 5 5 4 11 12 6 9 12 4 5 6 10 9 6 5 7 7 7 6 7 7 13 6 7 4 8
38:  4 3 5 5 5 7 5 5 7 10 9 5 5 6 8 10 6 4 12 5 9 7 6 7 7 10 5 6 7 15 7 4 7 7 7 6 3 10
39:  5 2 5 4 2 6 9 5 2 7 10 8 7 5 5 9 12 2 7 5 10 2 10 3 7 7 4 5 7 7 7 7 7 7 5 7 5 9 11
37:  3 4 2 5 6 5 3 7 8 5 8 9 5 8 12 1 4 5 11 7 5 7 7 3 7 13 6 4 6 7 12 7 7 7 7 6 8
38:  3 4 5 4 5 7 4 5 9 7 6 8 8 10 5 4 13 7 5 4 5 10 3 7 6 7 11 5 7 7 7 6 7 14 7 7 3 14
37:  4 3 4 3 5 5 5 5 7 8 10 5 5 5 12 9 10 8 5 8 5 7 6 6 5 11 5 12 5 7 7 7 7 7 7 7 10
37:  3 4 5 5 3 7 5 8 6 10 8 2 9 9 3 5 7 5 5 7 8 7 4 6 7 11 7 7 7 7 6 7 12 7 6 7 10
38:  5 2 4 4 5 4 5 7 5 9 7 5 3 9 8 10 6 9 5 4 10 8 6 6 7 7 7 7 7 7 7 13 6 6 6 5 7 10
3 early Golds followed by 2 early Provinces.  Overall, they all get 6-7 Provinces quite quickly, followed by long sequences of 7s until you get the last.

Amusingly, only one of those hits the worst-case scenario of not buying Gold until Turn 9.  Any way to pull those out for, say, T > 35, and see if that occurs in more cases?  Any cases where T[gold] < 9 are automatically not worst-case.

(Unless someone could explain why getting the first Gold at 9 is better than getting it at 8.)
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