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pitythefool

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Best buying strategy
« on: January 23, 2020, 01:31:06 am »
+7

Another pick-your-kingdom-and-maximize-your-VP problem.  But in this one, you only have one turn.  And you have just trashed the last card in your deck.  You have no cards in your hand, no cards set aside, no cards on a Tavern mat, no cards on a Native Village mat, no cards on any mat, no cards at all.  You have not purchased any projects or events.  But you do have one million coffers!  How will you spend them?

Pick any legal kingdom except it cannot contain Innovation, Villa, or Mission.  They would allow unbounded VP gains by allowing actions to be played.  All stacks in the kingdom are infinite and never run out of cards.  Split piles will have their normal contents repeated.  For example, the Catapult/Rocks pile would contain 5 Catapults, then 5 Rocks, then 5 Catapults, etc.  You can choose the makeup and order of the Ruins pile.  Assume solo play.

Clearly, Travelling Fair will help the task of buying many cards.  Canal is a good prospect as well.  Will you buy Colonies?  Dominate?

How about Obelisk on Forum?  No Travelling Fair needed since it comes with an extra buy.  You could buy 200,000 Forums for 400,000 VP.  Not so great.  Of course you are not limited to purchasing from a single pile and there are many non-victory cards you can add to your kingdom to help.

As an example, Iíll share the best strategy I came up with for purchasing Castles just to get you thinking and also so you donít have to analyze it yourself for comparison.

Action cards: Stonemason, Watchtower, Cultist, Castles, Forge
Landmarks: Tomb, Obelisk on Stonemason.
Events: Travelling Fair, Ferry, Alms, Ball, Salt the Earth, Triumph
Project: Canal

All purchases are proceeded with buying Travelling Fair.

Buy Canal.  (All cards are 1 coin cheaper.)
Buy Ferry for Stonemason.  (Ferries are free.)
(Ferry canít be applied to Castles since it is not an action supply pile.)
Buy Alms for a Watchtower.
Buy Salt the Earth, trash a Cultist, draw Watchtower into hand.
--- start of Castle buying loop ---
Buy Ball, gain Humble Castle and Crumbling Castle.  Topdeck both.
We will be topdecking all Castles, discarding all Stonemasons, and trashing everything else.
Next Castle in the rotation is Small Castle which is an action.
Buy Stonemason, overpay by 4, gain Small Castle and Cultist.
Reveal Watchtower, trash Small Castle, gain Haunted Castle which comes with a Gold.
(We trashed Small Castle, because itís on-gain effect is to gain the next Castle.  We effectively skip over it and rotate through the stacks faster, gaining more of the better Castles.)
Reveal Watchtower, trash Cultist, discard -1 card token from Ball, draw two Castles into hand.
Buy Stonemason, gain Opulent Castle and Forge.
Buy Sprawling Castle, gaining 3 Estates.
Buy Grand Castle.  This comes with a bonus of 1 VP per Castle in our hand.  We have two.  Next iteration, we will have four, etc.
Buy Kingís Castle.
--- repeat a total of 19,607 times ---
We have just enough money to do an additional partial iteration up to buying Opulent Castle, then use our last buy (without Travelling Fair) on Triumph which leaves us in debt.


We will have gained 137,253 Castles worth 294,105 basic VP.   Our 19,608 Humble Castles are worth 1 VP per Castle gained which is 2,691,256,824 VP.  19,607 Kingís Castles yield 2 VP per Castle for 5,382,239,142 VP.  Grand Castles yield bonus VP for Castles in hand which accumulated 384,454,056 VP.  The Stonemasons yielded 78,428 Obelisk VP.  Tomb also yielded 78,428 VP.  We gained 58,822 Estates.  And triumph gave us 313,724 VP for a grand total of 8,458,773,529 VP.

EDIT:  Oops.  Salt the Earth cannot trash a Cultist.  Instead, to get the Watchtower in hand, buy a Blessed Village and receive the Sea's Gift as a boon (+1 card).  This does not change the VP calculation.

EDIT 2:  Singletee has pointed out that I also made egregious errors with Ball and Small Castle.  I'm not going to recalculate the significant reduction of VP that this would cause.  Just ignore the example.  Thanks singletee.

This is not the best possible kingdom or strategy.  Letís see what you come up with.


« Last Edit: January 23, 2020, 09:34:19 am by pitythefool »
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singletee

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Re: Best buying strategy
« Reply #1 on: January 23, 2020, 04:58:36 am »
0

Interesting puzzle!

You have made an error with Small Castle, though. Small Castle doesn't have any on-gain or on-trash ability; it allows you to trash it to gain a Castle when played. Also, Ball gives the -$1 token, not the -1 Card token.
« Last Edit: January 23, 2020, 05:02:13 am by singletee »
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singletee

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Re: Best buying strategy
« Reply #2 on: January 23, 2020, 06:13:10 am »
+5

My first try, with some rough approximations:

Buy Watchtower and Blessed Village to get a Watchtower in hand. I'm not worried about how much this costs because it is very small compared to the rest.

Then, x times, buy Tfair and Banquet for Catacombs, trashing Catacombs for Feodum, trashing Feodum for Silvers.
Then y times, buy Tfair and Triumph.
The coin cost of this is 5x+7y, and it yields (1/2)y2+7xy VPs.
Substituting for x and rearranging, we get VPs = (7/5)cy - (93/10)y2. Here c (for "coffers") is equal to 1,000,000.
We want to maximize this, so take the derivative and set it to 0: (7/5)c - (93/5)y = 0
We then get y ~= 75268, and plug this back into the cost equation to get x ~= 94623.
Put these into the VP equation to get VPs ~= 52,687,223,660.
« Last Edit: January 23, 2020, 06:16:57 am by singletee »
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bitwise

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Re: Best buying strategy
« Reply #3 on: January 23, 2020, 01:29:24 pm »
+2

I'll copy your work with a bit of generalization.

To change the variables a bit to make it easier to recalculate when we can find different constants, let x be the amount of coin spent on buying the thing that generates card gain (Banquet in this case), and y be the amount of coin spent on Triumph.

The total number of cards gained before triumphing is x * (# cards gained per buy / $ cost per buy) = x * E. (I call it E for efficiency.) For TFair/Banquet -> Catacombs -> Feodum -> 3 Silver, that coefficient E is 7 / 5.

The total VP is (1/2)(y/7)^2+E*x*(y/7), to be maximized for x+y = C. The expression is 1/98 * (y^2 + 14*E*x*y) = 1/98 * (y^2+14*E*y*(C-y)) = 1/98 * ((1-14*E)y^2+14*E*C*y).
Ignoring the case where 14E < 1 (in which case it's best to just buy all Triumphs), that is maximized when its derivative is 0, or 2*(1-14E)y + 14*E*C = 0, or y = 7*E*C/(14E-1).

Plugging that back in gives 1/98 * ((1-14E)((7*E*C)/(14E-1))^2 + 14*E*C*(7*E*C)/(14E-1)) = 1/98 * ((7*E*C)^2/(1-14E) + 2*(7*E*C)^2/(14E-1)) = 1/98 * ((7*E*C)^2/(14E-1)) = 1/2 (E^2*C^2/(14E-1)) = 1/2 * C^2 * E^2/(14E-1).

I tried my formula with C = 1000000 and E = 7/5 and got 52,688,172,043 so I think the formula looks good.

Note that adding Border Village and Canal to the kingdom would allow us to have E = 8/5 for approximately 59,813,084,112 points. Also, if we include Stonemason and Ferry that, we can buy Stonemason overpaying by 5 instead of banquet. Each Border Village is 6 cards (BV, Catacombs, Feodum, 3 Silver), so each buy is 13 cards (SM + 2 BV), and costs 2 for the Travelling Fair and 5 for the Stonemason plus overpay, giving E = 13/7. This gives approximately 68979591836 points if I set C=1000000.

I guess now we are buying enough things as setup that I should count them. Canal costs 9, Ferry costs 5, Watchtower + BV costs 7 with Ball, and we start with 1000000 + 2 (our starting buy) + 1 (borrow), for C=999982. Then the formula gives approximately 68,977,108,593 VP.
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bitwise

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Re: Best buying strategy
« Reply #4 on: January 23, 2020, 01:41:23 pm »
+2

Oh, we can get way more efficient by making our buy Advance on a Fortress in our hand for Border Village, getting E = 6 / 2.

Kingdom actions needed:
Watchtower, Blessed Village, Fortress, Border Village, Catacombs, Feodum

Setup:
Buy Watchtower (5), Buy Blessed Village for +1 Card (6), Alms for Fortress and trash (2). (I had Ball before for Watchtower + Blessed Village but I'm not so sure that actually lets you draw the Watchtower?)
So C = 1000003 - 13 = 999990, and the approximate VP is 1/2 * (999990)^2 * 3^2/(14*3-1) = 109,753,902,450 VP.
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pitythefool

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Re: Best buying strategy
« Reply #5 on: January 23, 2020, 03:38:15 pm »
+2

Oh, we can get way more efficient by making our buy Advance on a Fortress in our hand for Border Village, getting E = 6 / 2.

So C = 1000003 - 13 = 999990, and the approximate VP is 1/2 * (999990)^2 * 3^2/(14*3-1) = 109,753,902,450 VP.
Well it appears that Triumph is very well named.  Good job all!  I had considered Triumph, but the "efficiency" I achieved was not high enough to beat my Duke/Duchy strategy.  You've done far better.
I'm going to summarize my strategy here anyway since it uses a mechanism not yet considered and it may spark more ideas.
Purchase Canal, Ferry on Stonemason, Ball for Watchtower and Blessed Village.
Then buy Stonemason, overpay by 1 to gain 2 Squires, trash them with Watchtower to gain 2 attack cards, repeat 333,326 times.
Finally, Alms for a Duchy
The attack gards we gain follow this sequence:   Gain 3 Vampires and topdeck them.  Every 4th attack card is a Cultist which we trash and draw the Vampires into our hand.
NIGHT PHASE
We can play our 499,989 Vampires to gain Duchies and Dukes for about 62.5*10^9 VP.
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pitythefool

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Re: Best buying strategy
« Reply #6 on: January 23, 2020, 05:34:17 pm »
+2

Oh, we can get way more efficient by making our buy Advance on a Fortress in our hand for Border Village, getting E = 6 / 2.

So C = 1000003 - 13 = 999990, and the approximate VP is 1/2 * (999990)^2 * 3^2/(14*3-1) = 109,753,902,450 VP.
Well it appears that Triumph is very well named.  Good job all!  I had considered Triumph, but the "efficiency" I achieved was not high enough to beat my Duke/Duchy strategy.  You've done far better.
I'm going to summarize my strategy here anyway since it uses a mechanism not yet considered and it may spark more ideas.
Purchase Canal, Ferry on Stonemason, Ball for Watchtower and Blessed Village.
Then buy Stonemason, overpay by 1 to gain 2 Squires, trash them with Watchtower to gain 2 attack cards, repeat 333,326 times.
Finally, Alms for a Duchy
The attack gards we gain follow this sequence:   Gain 3 Vampires and topdeck them.  Every 4th attack card is a Cultist which we trash and draw the Vampires into our hand.
NIGHT PHASE
We can play our 499,989 Vampires to gain Duchies and Dukes for about 62.5*10^9 VP.
A devious idea just popped into my head.
It's sort of a Night Phase north of the Artic Circle as it lasts nearly forever.

Previously we had ~500,000 Vampires to play in our Night Phase.
They can be exchanged for any card, up to 5 coins, except another Vampire.
Well, we can use one to gain a Squire, trash it with Watchtower, and gain a Vampire that way.
We can even stretch this out a bit more since we have Canal and Border Village.
Play Vampire, gain Border Village and Catacombs, trash Catacombs, gain Squire, trash Squire, gain Vampire and topdeck it.
Meanwhile, the original Vampire is exchanged for a Bat.
Every 4th iteration, substitute a Cultist, and draw the new Vampires into our hand.
Every iteration gains 5 cards.
If we continued doing this, our 500,000 Vampires could end up playing ~2,000,000 Vampires in total before being exhausted.
Here's the payoff.  Don't play Vampires all the way to extinction.  At some point, switch over and start gaining Monasteries.
We can play these to trash Fortress millions of times for big Tomb points.
I'd flesh it out more, but I'm off to work.

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bitwise

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Re: Best buying strategy
« Reply #7 on: January 23, 2020, 05:44:25 pm »
+2

I hadn't thought of a mega Night phase part of the turn--that's pretty interesting. Some thoughts about that:
  • Gaining a lot of Fortresses and Monasteries to hand would be another quadratic solution, though I think performs worse.
  • Monastery's trash can draw cards (Cultist), and gain Changelings, which can turn into more Monastery. Unfortunately, Changeling can't turn into any of the actions (we never play them) or Vampire (it gets exchanged).
  • Oh wait, there's an infinite combo with Vampire: play Vampire to gain Cultist, trashing it to draw 3 cards, and "gain" Bat (it goes to our discard but doesn't count as a gain for other effects) drawing 3 cards. Play Bat (trashing Fortress) to "gain" Vampire. This loop can repeat indefinitely, and nets a draw of 1 card every time. With Tomb, this is already infinite, but we can also use the extra card draw to sometimes not gain Cultist but gain other <= 5 costs instead and use those too. Heck, we can just gain Estates and draw them with the infinite extra draw + gains.

I also found an improvement to the Triumph solution that uses the fact that you can collect boons from Blessed Village to save for next turn to remove them from the Boon deck, and make it so that you get The Earth's Gift (discard a treasure to gain a <= 4 cost) every time you gain Blessed Village. I believe that it doesn't work to gain Blessed Village from this effect to get the boon again immediately, since the boon will stay out until you resolve all the effects, but you can still get some extra mileage out of it.

The basic loop is
A) Advance to gain BV (Border Village), Catacombs, bV (blessed Village), discard a Treasure to gain Catacombs, Squire, Cultist, which gains 6 cards and net draws 2 cards (discard a treasure, then +3 cards).
B) Advance to gain BV, Catacombs, bV, discard to gain Catacombs, Feodum, 3 Silvers, which gains 8 cards and net draws -1 card.

Then we can do A) B) B) in a cycle for an efficiency of 22/6 = 11/3.

Pessimistically assuming it costs 100 to do the setup, the total VP would be about 133,527,374,404 VP.
« Last Edit: January 23, 2020, 05:47:49 pm by bitwise »
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bitwise

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Re: Best buying strategy
« Reply #8 on: January 23, 2020, 11:11:04 pm »
0

Oh, didn't realize that Monastery didn't make you trash all the cards at the same time, letting it trash the same Fortress over and over. It definitely takes less than 7 money to gain a Monastery and draw it into hand. At the very least, we can Advance -> BV -> Catacombs -> Monastery 3 times and Advance -> BV -> Catacombs -> Squire -> Cultist 1 time to get 3 Monasteries in hand for 8 money. This is going to get a lot more VP, but I'll put off a calculation for now.
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pitythefool

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Re: Best buying strategy
« Reply #9 on: January 23, 2020, 11:54:52 pm »
+2

Oh, didn't realize that Monastery didn't make you trash all the cards at the same time, letting it trash the same Fortress over and over. It definitely takes less than 7 money to gain a Monastery and draw it into hand. At the very least, we can Advance -> BV -> Catacombs -> Monastery 3 times and Advance -> BV -> Catacombs -> Squire -> Cultist 1 time to get 3 Monasteries in hand for 8 money. This is going to get a lot more VP, but I'll put off a calculation for now.
Yes, I was surprised at your comment that it would perform worse.  Triumph is 1 VP per card gained.  Monastery/Tomb would also be 1 VP per card gained.  But there is the problem of getting the cards into the hand.  On the other hand, there is the usage of Vampires to gain more Vampires.

A simplistic calculation goes like this. (no recent optimizations added; all cards except Vampire and Monastery get trashed).
Buy Canal, Ball for Watchtower and Blessed Village, Alms for Fortress.  Only 19 coins.
Advance Fortress, gain Border Village and Catacombs; trash Catacombs, gain Squire; trash Squire, gain Vampire or Cultist (3 Vampires to 1 Cultist).  2 coins.
In the buy phase, we can gain 374,991 Vampires having gained 1,999,960 cards and trash 1,999,960 cards.
The night phase is similar.
Play Vampire, gain Border Village, Bat, and Catacombs; trash Catacombs, gain Squire; trash Squire, gain Vampire or Cultist or Monastery.
For every four Vampires we play, we end up with three more in hand, having gained another 20 cards and trashed another 17.
So we can do 93,747 iterations, bringing the total gained to 3,874,900 and total trashed to 3,593,659 with 281,241 new Vampires in hand.
We do 70,310 more iterations, bringing the total gained to 5,281,100 and total trashed to 4,788,929 with 210,930 Monasteries in hand.
Playing all of these allows us to trash the Fortress 5,281,100 * 210,930 times to bring the total Tomb VP points to 1.114*10^12.  This is not the optimal switchover point.  I just guessed.
So overall, the scores are on par to the Triumph method.

Quote from: bitwise
    Oh wait, there's an infinite combo with Vampire: play Vampire to gain Cultist, trashing it to draw 3 cards, and "gain" Bat (it goes to our discard but doesn't count as a gain for other effects) drawing 3 cards. Play Bat (trashing Fortress) to "gain" Vampire. This loop can repeat indefinitely, and nets a draw of 1 card every time. With Tomb, this is already infinite, but we can also use the extra card draw to sometimes not gain Cultist but gain other <= 5 costs instead and use those too. Heck, we can just gain Estates and draw them with the infinite extra draw + gains.
When I started this puzzle, I never imagined that a Night Phase engine could be created.  That you found an unbounded one is amazing.  It is sort of a spoiler for the puzzle now.  I don't think further work on it is worthwhile.

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pitythefool

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Re: Best buying strategy
« Reply #10 on: January 24, 2020, 12:34:52 am »
0


A simplistic calculation goes like this. (no recent optimizations added; all cards except Vampire and Monastery get trashed).
Buy Canal, Ball for Watchtower and Blessed Village, Alms for Fortress.  Only 19 coins.
Advance Fortress, gain Border Village and Catacombs; trash Catacombs, gain Squire; trash Squire, gain Vampire or Cultist (3 Vampires to 1 Cultist).  2 coins.
In the buy phase, we can gain 374,991 Vampires having gained 1,999,960 cards and trash 1,999,960 cards.
The night phase is similar.
Play Vampire, gain Border Village, Bat, and Catacombs; trash Catacombs, gain Squire; trash Squire, gain Vampire or Cultist or Monastery.
For every four Vampires we play, we end up with three more in hand, having gained another 20 cards and trashed another 17.
So we can do 93,747 iterations, bringing the total gained to 3,874,900 and total trashed to 3,593,659 with 281,241 new Vampires in hand.
We do 70,310 more iterations, bringing the total gained to 5,281,100 and total trashed to 4,788,929 with 210,930 Monasteries in hand.
Playing all of these allows us to trash the Fortress 5,281,100 * 210,930 times to bring the total Tomb VP points to 1.114*10^12.  This is not the optimal switchover point.  I just guessed.
So overall, the scores are on par to the Triumph method.
Whoa!  My first engine used Stonemason/Squire to gain Vampires for 3 Vampires per 6 coins.
For this last calculation I switched over to Advance/Fortress because I thought it was simpler and the same cost (and trashed more cards).  But it gains 3 Vampires per 8 coins.  I should have 33% more Vampires at the end of the buy phase.  With quadratic increases, that should increase my score by about 1.333^2 = 1.7777.  So I think it may have been better overall.  (But still trumped by unbounded).
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bitwise

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Re: Best buying strategy
« Reply #11 on: January 24, 2020, 12:44:27 am »
+1

Quote from: bitwise
    Oh wait, there's an infinite combo with Vampire: play Vampire to gain Cultist, trashing it to draw 3 cards, and "gain" Bat (it goes to our discard but doesn't count as a gain for other effects) drawing 3 cards. Play Bat (trashing Fortress) to "gain" Vampire. This loop can repeat indefinitely, and nets a draw of 1 card every time. With Tomb, this is already infinite, but we can also use the extra card draw to sometimes not gain Cultist but gain other <= 5 costs instead and use those too. Heck, we can just gain Estates and draw them with the infinite extra draw + gains.
When I started this puzzle, I never imagined that a Night Phase engine could be created.  That you found an unbounded one is amazing.  It is sort of a spoiler for the puzzle now.  I don't think further work on it is worthwhile.
I'm glad there's something infinite!. I think it's still reasonable to think about the puzzle where we restrict it to kingdoms that can't have an infinite score. As Cultist + Vampire is infinite, and Vampire seems pretty bad without Cultist existing, effectively we just need to ban Vampire.

Monastery can just be a straight replacement for the Triumph solution where we replace Triumph buys with buys that gain and draw Monastery. No Vampires are needed. Comparing buying 1 Triumph for 7 and gaining and drawing Monastery for 8/3 as a unit, it should be quite a big improvement because the Monastery gains more than 1 card, all the gains additionally count for all the Monasteries (whereas for Triumphs, the gained Estates only count for the current and future Triumphs), and it costs about 38% as much.
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trivialknot

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Re: Best buying strategy
« Reply #12 on: January 25, 2020, 12:14:43 am »
+3

Ignoring the infinite solutions, I think there's a solution that scales better than N^2.  You can buy a bunch of Tunnels, followed by a bunch of Cursed Villages.  Not sure how much you're allowed to stack hexes, but War should allow you to gain a bunch of gold (using Canal to skip past tunnels and Watchtower to trash Cursed Villages.)  Then buy a bunch of triumphs.  This doesn't sound very efficient, but it should scale as N^3, so with enough coffers it should eventually be better.
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trivialknot

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Re: Best buying strategy
« Reply #13 on: January 25, 2020, 01:10:41 am »
+1

Okay, here are some cost calculations:

9 for Canal (all these prices include TFair)
4 for Watchtower
5 for Blessed Village
3.5x for x Tunnels (using Ball)
3.5y for y Cursed Villages (using Ball, trashing all gains with Watchtower)
7z for z Triumphs

With every 12 Cursed villages, there are 15+x gains (Cursed Villages, Greed, Plague, Locusts, War).  Just put Locusts after Greed so you never trash a Tunnel.

So the total VP is:

2x for Tunnels
-4 for Twice Miserable
y/12*(15+x) for trashing (Tomb)
z for Estates
~z*(z/2+y/12*(15+x)) for Triumphs

I'm not going to solve for these variables exactly, but by far the most important term is proportional to x*y*z, and that can be maximized by splitting money equally between Tunnels, Cursed Villages, and Triumphs.  I estimate about 35,998,000,000,000 VP, which improves on previous solutions by two orders of magnitude.
« Last Edit: January 25, 2020, 01:13:56 am by trivialknot »
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bitwise

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Re: Best buying strategy
« Reply #14 on: January 25, 2020, 02:01:06 am »
0

Wow, really nice find! Glad I put off my calculations.  :D

This solution can also take in a lot of the improvements we had for the N^2 solution--Ball can be replaced with Advance on Fortress, and Triumph can be replaced with Monastery and Cultist to draw them.

The kingdom would need
Watchtower, Blessed Village, Tunnel, Cursed Village, Fortress, Monastery, Cultist.
This leaves enough room to add Border Village and Catacombs for some additional value.

I think that the thing with triggering the discard treasure for 4 cost boon isn't going to be worth it in this solution, so that's a little less to calculate. Also, one thing with having Monastery Tomb be the point source is that we can add Sewers to approximately double our points. I didn't check your 35*10^12 estimate, but assuming that's right, I'm hopeful that this would give over 10^15 VP.
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pitythefool

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Re: Best buying strategy
« Reply #15 on: January 25, 2020, 02:55:13 am »
0

Ignoring the infinite solutions, I think there's a solution that scales better than N^2.  You can buy a bunch of Tunnels, followed by a bunch of Cursed Villages.  Not sure how much you're allowed to stack hexes, but War should allow you to gain a bunch of gold (using Canal to skip past tunnels and Watchtower to trash Cursed Villages.)  Then buy a bunch of triumphs.  This doesn't sound very efficient, but it should scale as N^3, so with enough coffers it should eventually be better.
I like the idea.  The use of War seems problematic.
There are 12 Hexes.  The stack is not infinite.  You're allowed to choose the shuffle order, but you have to slog through all of them.  (Even if you interpreted it to be an infinite stack, it would be 12, followed by another 12, etc.).  That means only 1 in 12 Cursed Villages will trigger War.

Might I suggest trying it in the Night Phase?  Ghost will nicely discard the Tunnels and Monastery is a great replacement for Triumph.
With Vampire, this would be unbounded.  Even without Vampire, the night cards can give the strategy a boost.

I'd stock up on a lot of Catacombs, keep Changelings whenever possible, then buy a few Exorcists and Monasteries.
Before proceeding to the night phase, put Ferry on Border Village.
Then play an Exorcist, trash a Catacombs gaining a Ghost, then gaining a Border Village, and another Exorcist card costing less than 4 coins, maybe a Monastery.  Exchange the Border Village for a Changeling.
To draw them in, play an Exorcist, trash a Catacombs gaining a Ghost, then gaining a Border Village, and a Squire.
Trash the Squire with Watchtower, gain a Cultist, trash that with Watchtower, draw 3 cards.  Exchange the Border Village for another Changeling.
Play the Monasteries last to trash Fortress and gain Tomb points.
The Changelings can gain you more copies of Exorcist and Monastery.
I haven't worked out the ratios or best order, but I think this will do better.
« Last Edit: January 25, 2020, 03:21:42 am by pitythefool »
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pitythefool

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Re: Best buying strategy
« Reply #16 on: January 25, 2020, 10:47:09 am »
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Ignoring the infinite solutions, I think there's a solution that scales better than N^2.  You can buy a bunch of Tunnels, followed by a bunch of Cursed Villages.  Not sure how much you're allowed to stack hexes, but War should allow you to gain a bunch of gold (using Canal to skip past tunnels and Watchtower to trash Cursed Villages.)  Then buy a bunch of triumphs.  This doesn't sound very efficient, but it should scale as N^3, so with enough coffers it should eventually be better.
I like the idea.  The use of War seems problematic.
There are 12 Hexes.  The stack is not infinite.  You're allowed to choose the shuffle order, but you have to slog through all of them.  (Even if you interpreted it to be an infinite stack, it would be 12, followed by another 12, etc.).  That means only 1 in 12 Cursed Villages will trigger War.

Might I suggest trying it in the Night Phase?  Ghost will nicely discard the Tunnels and Monastery is a great replacement for Triumph.
With Vampire, this would be unbounded.  Even without Vampire, the night cards can give the strategy a boost.

I'd stock up on a lot of Catacombs, keep Changelings whenever possible, then buy a few Exorcists and Monasteries.
Before proceeding to the night phase, put Ferry on Border Village.
Then play an Exorcist, trash a Catacombs gaining a Ghost, then gaining a Border Village, and another Exorcist card costing less than 4 coins, maybe a Monastery.  Exchange the Border Village for a Changeling.
To draw them in, play an Exorcist, trash a Catacombs gaining a Ghost, then gaining a Border Village, and a Squire.
Trash the Squire with Watchtower, gain a Cultist, trash that with Watchtower, draw 3 cards.  Exchange the Border Village for another Changeling.
Play the Monasteries last to trash Fortress and gain Tomb points.
The Changelings can gain you more copies of Exorcist and Monastery.
I haven't worked out the ratios or best order, but I think this will do better.
Okay.  First, let it be known that I've been getting little sleep.  I was up again most of the night thinking about this.  Puzzles like this prevent me from turning off my brain.  Who the hell posted this infernal ... Oh, wait.

So I've been thinking that Monasteries are probably cheaper to gain and draw in the buy phase, so I'll skip discussing those.
Another tweak I've been thinking about is deck management.
Edit:  Complete nonsense follows; ignore it.
After playing a Ghost, your entire deck will be in the discard pile.  If you immediately play another Ghost, you have no draw pile, your discards will be shuffled and your entire deck will then be scanned and discarded.  So you only get to discard all the Tunnels once.  It doesn't help to topdeck some gained Gold, they'd just be discarded, then the deck shuffled and scanned once.  But if you force a draw, your deck will become your draw pile and a Ghost will scan it twice.  You could  achieve this by overdrawing with a Cultist, i.e. topdeck 2 cards, draw 3.  But this is the waste of a card draw and you might actually draw a Tunnel.  Still helpful, but better is Night Watchman.  It's also cheaper and gained to the hand.

So after some initial setup, the loop would be
Play an Exorcist, trash a Catacombs gaining a Ghost, then gaining a Border Village, and a Night Watchman (to hand).  Exchange the Border Village for a Changeling.
Play an Exorcist, trash a Catacombs gaining a Ghost, then gaining a Squire.  Trash the Squire with Watchtower, gain a Cultist, trash that, draw 2 Ghosts and a Changeling.
Play Changeling, gain a Night Watchman to hand.
Play Night Watchman to load the draw pile.
Play Ghost to scan your deck twice.
Play Night Watchman to load the draw pile.
Play Ghost to scan your deck twice.

So for 2 Exorcists and 2 Catacombs, we've done some extra gaining and scanned the deck four times.
« Last Edit: January 25, 2020, 06:08:37 pm by pitythefool »
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bitwise

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Re: Best buying strategy
« Reply #17 on: January 25, 2020, 11:53:37 am »
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Great improvement again!

I'm not sure I understand what you're saying about Ghost going through a deck twice. From my understanding, a single Ghost can't reveal any card more than once, because when it reveals cards, it temporarily sets them aside so that they're not shuffled if the deck needs to be reshuffled.
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pitythefool

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Re: Best buying strategy
« Reply #18 on: January 25, 2020, 05:23:32 pm »
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Great improvement again!

I'm not sure I understand what you're saying about Ghost going through a deck twice. From my understanding, a single Ghost can't reveal any card more than once, because when it reveals cards, it temporarily sets them aside so that they're not shuffled if the deck needs to be reshuffled.
You're quite right.  Once again, my error.  So Night Watchman is useless.
« Last Edit: January 25, 2020, 05:26:20 pm by pitythefool »
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pitythefool

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Re: Best buying strategy
« Reply #19 on: January 25, 2020, 10:38:01 pm »
+1

Ignoring the infinite solutions, I think there's a solution that scales better than N^2.  You can buy a bunch of Tunnels, followed by a bunch of Cursed Villages.  Not sure how much you're allowed to stack hexes, but War should allow you to gain a bunch of gold (using Canal to skip past tunnels and Watchtower to trash Cursed Villages.)  Then buy a bunch of triumphs.  This doesn't sound very efficient, but it should scale as N^3, so with enough coffers it should eventually be better.

Here's how I think it looks so far.

Squire, Watchtower, Fortress, Tunnel, Catacombs, Cultist, Border Village, Changeling, Exorcist, Monastery.
Alms, Ball, Advance
Tomb

Each buy is proceeded with Travelling Fair, all buys (not trashed) will be topdecked, except Tunnels.

Buy Ball for Watchtower and Blessed Village, gain Sea's Gift, draw Watchtower
Buy Alms for Fortress, reveal Watchtower, trash it.
3 cards gained; 999,990 coins remaining.

We will now be purchasing Tunnels, Catacombs, Exorcists, and Monasteries.
Each with the same method for 2 coins each.

Play Advance on Fortress, gain Border Village and Catacombs.  After resolving on-gain for Border Village, exchange it for a Changeling and trash with Watchtower.
   Either keep Catacombs or reveal Watchtower, trash it, and gain one of {Tunnel, Exorcist, or Monastery}.

The process of gaining 1 Catacombs gains 3 cards total.
Gaining 1 Tunnel, Exorcist, or Monastery gains 4 cards each.

Now we have to draw all the Catacombs, Exorcists, and Monasteries into our hand.

Play Advance on Fortress, gain Border Village, Catacombs, and Changeling. Trash Catacombs, gain Squire. Trash Squire, gain Cultist.  Trash Cultist, draw 3 cards.  Trash Changeling.

This process for drawing 3 cards costs 2 coins and gains 5 cards.

Night Phase

Play Exorcist, trash Catacombs, gain Ghost, Gain Exorcist
Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw 2 Ghosts and an Exorcist.

1 Exorcist and 2 Catacombs are combined into 2 Ghosts, gaining a total of 5 cards in the process.

Play all Ghosts
Play all Monasteries, trashing Fortress       

---- breaking down the numbers ----

Buying 166,654 Tunnels costs 333,308 coins and gains 666,616 cards.
Buying 41,666 Exorcists costs 83,332 coins and gains 166,614 cards.
Buying 83,330 Catacombs costs 166,660 coins and gains 249,990 cards.
Buying 125,009 Monasteries costs 250,018 coins and gains 500,036 cards.
Drawing 41,666 Exorcists + 83,330 Catacombs + 125,009 Monasteries takes 83,335 Cultists.  Cost 166,670 coins gaining 416,675 cards.
We have exactly 2 coins left.  Buy Travelling Fair, gain a Tunnel (as above) and then buy Triumph.
The Estate gave us a total of 1,999,936 gained cards.  That's our score so far.
Night Phase
Combining our 41,666 Exorcists and 83,330 Catacombs into 83,330 Ghosts and drawing them gains another 208,325 cards.
Playing each Ghost gains another 166,655 Gold, which brings our total card gain to 13,889,361,086.
Playing each Monastery gains that many more VP, for a total score of 1.7362951e+15 VP.

We started with one million coins and effectively gained 1.7 billion VP per coin.

I want to thank everybody who helped get us this far.  IMHO, collaboration is highly underrated in the real world.

And now that I'm all done with this analysis and have written it up, I just realized that Sewers would practically double the total VP.
« Last Edit: January 25, 2020, 10:41:59 pm by pitythefool »
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bitwise

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Re: Best buying strategy
« Reply #20 on: January 26, 2020, 02:27:20 am »
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I believe that exchanging for Changeling doesn't count as an additional gain, meaning the Catacombs gain gets 2 cards total and the Tunnel or Exorcist or Monastery gain gets 3 cards, lowering the point totals.

On the other hand, we could consider keeping a Changeling sometimes, since they can be turned into Exorcists or Monasteries. On the margin, we save an Advance to gain the Exorcist/Monastery, and have to pay the cost of drawing the card again in the Night phase. Drawing the card again in Night phase costs 1/3 of an Exorcist-Catacombs in hand pair. That Exorcist-Catacombs in hand cost 2 + 2/3 Advances in the first place, so the total cost of that draw would be 8/9 Advances, so this should be a net savings.
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pitythefool

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Re: Best buying strategy
« Reply #21 on: January 26, 2020, 02:59:08 am »
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I believe that exchanging for Changeling doesn't count as an additional gain, meaning the Catacombs gain gets 2 cards total and the Tunnel or Exorcist or Monastery gain gets 3 cards, lowering the point totals.
So it's like Trader then, the on-gain effects happen for the card, but then you don't actually gain it.  That makes sense.  That would reduce the VP, but not significantly.
Quote
On the other hand, we could consider keeping a Changeling sometimes, since they can be turned into Exorcists or Monasteries. On the margin, we save an Advance to gain the Exorcist/Monastery, and have to pay the cost of drawing the card again in the Night phase. Drawing the card again in Night phase costs 1/3 of an Exorcist-Catacombs in hand pair. That Exorcist-Catacombs in hand cost 2 + 2/3 Advances in the first place, so the total cost of that draw would be 8/9 Advances, so this should be a net savings.
Would it be?  I'm too tired to think about it now, but it's 1/3 of a Cultist draw to gain the Changeling in hand, then 1/3 of another Cultist draw to draw the gained card.  I'll think about it when I finally get another good night of sleep.  It's 3 AM locally as I write this.

Some other mistakes I made.  My program was off by one on the Tunnel count, so I unexpectedly had 2 coins left.  After buying another Tunnel, I had a free buy but no coins.  Triumph was an insignificant purchase.  I could have bought another Tunnel.

I was so focused on card costs, the card gain count, and the sequence of things that I did not track card trashing (other than for the Monasteries).  That would add up considering we could trash all the gained Gold.

And of course those Tunnels are worth some VP just for being Tunnels.

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Re: Best buying strategy
« Reply #22 on: January 26, 2020, 10:49:00 am »
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I believe that exchanging for Changeling doesn't count as an additional gain, meaning the Catacombs gain gets 2 cards total and the Tunnel or Exorcist or Monastery gain gets 3 cards, lowering the point totals.
So it's like Trader then, the on-gain effects happen for the card, but then you don't actually gain it.  That makes sense.  That would reduce the VP, but not significantly.

Not quite. For these purposes, it's the same, but in reality there is a difference:
With Trader, it's the other card which doesn't count as a gain, and the Silver which does.
With Changeling, it's the other card which counts as a gain, and the Changeling itself which doesn't.
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pitythefool

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Re: Best buying strategy
« Reply #23 on: January 26, 2020, 01:04:43 pm »
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Quote
On the other hand, we could consider keeping a Changeling sometimes, since they can be turned into Exorcists or Monasteries. On the margin, we save an Advance to gain the Exorcist/Monastery, and have to pay the cost of drawing the card again in the Night phase. Drawing the card again in Night phase costs 1/3 of an Exorcist-Catacombs in hand pair. That Exorcist-Catacombs in hand cost 2 + 2/3 Advances in the first place, so the total cost of that draw would be 8/9 Advances, so this should be a net savings.
Would it be?  I'm too tired to think about it now, but it's 1/3 of a Cultist draw to gain the Changeling in hand, then 1/3 of another Cultist draw to draw the gained card.  I'll think about it when I finally get another good night of sleep.  It's 3 AM locally as I write this.
You're right of course.  When buying Catacombs, we can keep every other Changeling.  Then buy the Monasteries and Exorcists as before, trashing their Border Villages.  Trash enough Cultists to draw the Exorcists and Monasteries, but then enter Night Phase with the rest still on the deck.

Night phase play then starts out with this loop:

Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and a Changeling.
Play Changeling on Exorcist.
Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and an Exorcist.

We save the cost of drawing all of the Catacombs into our hand.  Big win.

(This replaces an earlier post where I miscalculated the number of Exorcists required.  We still need one for every two Catacombs, plus 1.)
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pitythefool

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Re: Best buying strategy
« Reply #24 on: January 26, 2020, 05:03:24 pm »
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Quote
On the other hand, we could consider keeping a Changeling sometimes, since they can be turned into Exorcists or Monasteries. On the margin, we save an Advance to gain the Exorcist/Monastery, and have to pay the cost of drawing the card again in the Night phase. Drawing the card again in Night phase costs 1/3 of an Exorcist-Catacombs in hand pair. That Exorcist-Catacombs in hand cost 2 + 2/3 Advances in the first place, so the total cost of that draw would be 8/9 Advances, so this should be a net savings.
Would it be?  I'm too tired to think about it now, but it's 1/3 of a Cultist draw to gain the Changeling in hand, then 1/3 of another Cultist draw to draw the gained card.  I'll think about it when I finally get another good night of sleep.  It's 3 AM locally as I write this.
You're right of course.  When buying Catacombs, we can keep every other Changeling.  Then buy the Monasteries and Exorcists as before, trashing their Border Villages.  Trash enough Cultists to draw the Exorcists and Monasteries, but then enter Night Phase with the rest still on the deck.

Night phase play then starts out with this loop:

Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and a Changeling.
Play Changeling on Exorcist.
Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and an Exorcist.

We save the cost of drawing all of the Catacombs into our hand.  Big win.

(This replaces an earlier post where I miscalculated the number of Exorcists required.  We still need one for every two Catacombs, plus 1.)
I've been pondering this a great deal.

The old method

  Play Exorcist, trash Catacombs, gain Ghost, Gain Exorcist
  Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw 2 Ghosts and an Exorcist.

The altered method

  Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and a Changeling.
  Play Changeling on Exorcist.
  Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and an Exorcist.

Both gain the same amount of Ghosts and consume the same amount of Exorcists and Catacombs. (1 to 2 ratio)
The 1st loop creates two Ghosts and an Exorcist and draws them. 3 cards drawn per iteration.
The 2nd loop creates two Ghosts and an Exorcist and draws them plus 2 Catacombs.  Effectively 5 cards drawn per iteration.
In both cases, we did not have to buy the Exorcist, we created it and drew it.
The win in the second case was that we got all of the Catacombs into our hand for free.
Though the 2nd loop actually drew 6 cards per loop, it only effectively drew 5 since we had to draw a Changeling to make the Exorcist.

We can do better if we draw the Changelings into our hand before the Night phase.  It is cheaper than drawing them in the Night phase. That way we get full efficiency from the Night phase draw.  We can no longer use the Changelings gained from buying the Catacombs because we can't get the right deck ordering.  But we will have more Monasteries than Catacombs.
If N is the number of Ghosts we want and M is the number of Monasteries, this is the order to buy.

Buy 2 Monasteries, no Changelings.  (Just wraps up the draw nicely)
Buy N-1 Catacombs, no Changelings.
Buy 1 Exorcist, no Changeling.
Buy 1 Catacombs, no Changelings.
Buy 1 Exorcist, no Changeling.
Buy M-N Monasteries, no Changelings.
Buy N-2 Monasteries, with Changelings.
Use Cultists to draw all the top Monasteries, Changelings, and 1 Exorcist, and 1 Catacombs.

Night phase starts with:

Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and an Exorcist.
-- repeat this loop N-2 times
   Play Changeling on Exorcist.
   Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, Catacombs, and an Exorcist.
-- end loop
Play Exorcist, trash Catacombs, gain Ghost, Gain Squire, trash it, gain Cultist, trash it, draw Ghost, and 2 Monasteries.

Instead of buying and drawing N/2 Exorcists into our hand before the Night phase, we are drawing N free Changelings into our hand before the Night phase.  Since a draw is cheaper than a buy, it's another win.  You were really onto something.

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