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Author Topic: Can you score exponentially many points ?  (Read 7345 times)

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gamesou

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Can you score exponentially many points ?
« on: November 10, 2011, 03:12:48 am »
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Let's try to think how the game would change if the number of cards in each pile was much larger than 10.

You play now Dominion with 10^42 cards in each pile. Let's say also that all possible kingdom cards are available.

How would you play ? Assuming no interference with opponents, there is a strategy that crushes every possible combo I can imagine.
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Davio

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Re: Can you score exponentially many points ?
« Reply #1 on: November 10, 2011, 03:26:31 am »
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There's the infinite Bishop with just a simple Plat-Plat-Bishop-Colony-RandomCard you can trash the Colony every turn for 11/2 + 1 = 6 VP and you have enough cash to buy a new Colony so you can do the same the next turn. The original solution has Gold-Gold-Silver-Bishop-Province for a result of 5 VP every turn.

If you want to extend the solution by including something like Monument, you could run into bad shuffle luck.
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Jimmmmm

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Re: Can you score exponentially many points ?
« Reply #2 on: November 10, 2011, 03:37:04 am »
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5 King's Courts + 5 Schemes + 4 Goons -> buy two Colonies, an Estate and 10 Coppers for 73 VP/turn. Add more KC + S + G for more.
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DStu

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Re: Can you score exponentially many points ?
« Reply #3 on: November 10, 2011, 03:44:55 am »
+1

Given so many time, I think it is best to assume perfect shuffle luck and try to create a KC-Bridge-IW-Watchtower-Great Hall/Nobles thing which can gain arbritrary many Provinces in one turn. Say you need 10 cards in your deck, that's a lot less than 10!=3.6x10^6 different hands, so that will trigger long before anybody can grap 10^42 cards normally.
Once you have that of course we have to find how we can make that happen most fastly (asymptotically perhaps).

Edit:
Say you have reduced evereything to $0 by Highways/Bridges, you can play KC/KC/IW/IW/CR to gain X/KC/KC/IW/IW/CR in hand, I think it should be possible to grind your way upwards from there...
« Last Edit: November 10, 2011, 03:53:57 am by DStu »
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gamesou

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Re: Can you score exponentially many points ?
« Reply #4 on: November 10, 2011, 04:03:30 am »
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Given so many time, I think it is best to assume perfect shuffle luck and try to create a KC-Bridge-IW-Watchtower-Great Hall/Nobles thing which can gain arbritrary many Provinces in one turn. Say you need 10 cards in your deck, that's a lot less than 10!=3.6x10^6 different hands, so that will trigger long before anybody can grap 10^42 cards normally.
Once you have that of course we have to find how we can make that happen most fastly (asymptotically perhaps).

Edit:
Say you have reduced evereything to $0 by Highways/Bridges, you can play KC/KC/IW/IW/CR to gain X/KC/KC/IW/IW/CR in hand, I think it should be possible to grind your way upwards from there...

AMAZING ! I never imagined that one could do such a thing !

My solution is completely different, and is crushed by yours. If there were "only" 10^8 cards in the supply, my solution would be competitive, but 10^42 is too close to infinity ...
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DStu

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Re: Can you score exponentially many points ?
« Reply #5 on: November 10, 2011, 04:18:03 am »
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Given so many time, I think it is best to assume perfect shuffle luck and try to create a KC-Bridge-IW-Watchtower-Great Hall/Nobles thing which can gain arbritrary many Provinces in one turn. Say you need 10 cards in your deck, that's a lot less than 10!=3.6x10^6 different hands, so that will trigger long before anybody can grap 10^42 cards normally.
Once you have that of course we have to find how we can make that happen most fastly (asymptotically perhaps).

Edit:
Say you have reduced evereything to $0 by Highways/Bridges, you can play KC/KC/IW/IW/CR to gain X/KC/KC/IW/IW/CR in hand, I think it should be possible to grind your way upwards from there...

AMAZING ! I never imagined that one could do such a thing !

My solution is completely different, and is crushed by yours. If there were "only" 10^8 cards in the supply, my solution would be competitive, but 10^42 is too close to infinity ...

I'm not sure that it works. But there is the "8 Colonies in 5 turns" puzzle on the blog, maybe one can modify it.
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jimjam

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Re: Can you score exponentially many points ?
« Reply #6 on: November 10, 2011, 05:47:30 am »
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If we're considering one-turn where the variable is the supply limit:

KC/Gainers is the best but will run out eventually. Therefore it's prudent to go Duke/Duchy first (quadratic *1/2 per green * fraction of greens (probably 1/2)). Vineyards are nice but harder to gain, so next would probably be Silk Roads. After that is Gardens. Anything else is basically worthless, really.
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DStu

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Re: Can you score exponentially many points ?
« Reply #7 on: November 10, 2011, 06:12:43 am »
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KC/KC/IW/IW/CR->KC/KC/IW/IW/CR/X needs 2CR and 2 IW to produce 3 buys, more actions than you need and 1X.
As whoever pulls something like that first will win, no matter if he buys Provinces or Duchy/Duke, it should not really matter how many points you have. But if you want to maximize them, too, with this method you get 2xN-const. many cards (you need 2 IW/KC's, but you get 4 gains each time you apply it). So it should be possible to finish (at least more or less) the Duchies and Dukes. Getting additional Silk Roads might get complicated, but I think it is possible throwing in TR for example to save IWs that can gain additional cards.

Problem is, if you just can finish Duchy/Duke and not the SR, you are stuck with only 2 empty piles, and a really bad deck, and if you're opponent has enough time and does not want to surrender, you must grind your way through the last pile. If you just finish the Provinces, you're done.

Edit: OK, it's not that bad, it's also full of KCs and IW. That should be enough to finish the last pile in about 10^42 turns...

Edit2: As X can also be a Potion, and you have enough buys, it should also be possible to get the Vineyards with this, which should be quite valuable because you have ~2.5x10^42 actioncards. I doubt that you get the SR to this, because that would mean to finish nearly 4 victorypiles.
« Last Edit: November 10, 2011, 06:23:30 am by DStu »
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philosophyguy

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Re: Can you score exponentially many points ?
« Reply #8 on: November 10, 2011, 07:34:27 am »
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Just a reminder that, with Scheme, shuffle luck is no longer an issue. You can always make sure you have the cards to draw your deck on top of your deck.
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DG

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Re: Can you score exponentially many points ?
« Reply #9 on: November 10, 2011, 08:52:15 am »
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I think Donald X. once mentioned something about infinite cards gaining and how it wasn't possible, at that time. I suspect that with some highway, king's court, ironworks combinations it would be possible. It doesn't really matter though as the supply piles are finite (practically, you're not going to ambassador cards to the supply just to gain them again).
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mathguy

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Re: Can you score exponentially many points ?
« Reply #10 on: November 10, 2011, 09:07:28 am »
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... (practically, you're not going to ambassador cards to the supply just to gain them again).

...In the thread about how can we pull off exponentially many KC-KC-Bridges.  ;)
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gamesou

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Re: Can you score exponentially many points ?
« Reply #11 on: November 10, 2011, 09:39:49 am »
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I was just trying my strategy on a usual board in a solitaire isotropic game. I don't think I was exceptionnally lucky, and I ended with 217 points in 14 turns.
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tlloyd

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Re: Can you score exponentially many points ?
« Reply #12 on: November 10, 2011, 10:29:07 am »
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You don't need more cards in play to score an infinite amount of points (more precisely, to continue accumulating point indefinitely): village, goons, ambassador(return two copper), buy two copper. Repeat. Obviously you can ramp this combo up to gain more points per turn, but this simple combo can continue gaining VP forever.
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DStu

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Re: Can you score exponentially many points ?
« Reply #13 on: November 10, 2011, 10:45:31 am »
+1

Quote
You don't need more cards in play to score an infinite amount of points (more precisely, to continue accumulating point indefinitely): village, goons, ambassador(return two copper), buy two copper. Repeat. Obviously you can ramp this combo up to gain more points per turn, but this simple combo can continue gaining VP forever.
Or: ||:Monument:||
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GendoIkari

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Re: Can you score exponentially many points ?
« Reply #14 on: November 10, 2011, 10:46:13 am »
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You don't need more cards in play to score an infinite amount of points (more precisely, to continue accumulating point indefinitely): village, goons, ambassador(return two copper), buy two copper. Repeat. Obviously you can ramp this combo up to gain more points per turn, but this simple combo can continue gaining VP forever.

Monument -> Buy nothing does the same; and simpler. ;)

*Edit* Ninja'd by 1 minute!
« Last Edit: November 10, 2011, 10:49:19 am by GendoIkari »
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theory

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Re: Can you score exponentially many points ?
« Reply #15 on: November 10, 2011, 11:32:47 am »
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You could just rephrase this puzzle (as with many other puzzles) to explicitly exclude points from VP chips.
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DStu

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Re: Can you score exponentially many points ?
« Reply #16 on: November 10, 2011, 12:48:47 pm »
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You could just rephrase this puzzle (as with many other puzzles) to explicitly exclude points from VP chips.

I'm not sure that is necessary, because when the task is to win a 10^42-card pile game asap, I'm quite confident that you won't need VP chips.
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Kirian

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Re: Can you score exponentially many points ?
« Reply #17 on: November 10, 2011, 01:01:36 pm »
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Problem is, if you just can finish Duchy/Duke and not the SR, you are stuck with only 2 empty piles, and a really bad deck, and if you're opponent has enough time and does not want to surrender, you must grind your way through the last pile. If you just finish the Provinces, you're done.

Edit: OK, it's not that bad, it's also full of KCs and IW. That should be enough to finish the last pile in about 10^42 turns...

The way you've set this up, though, you're also running out the KC and IW piles, right?  This ends the game on that turn.
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tlloyd

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Re: Can you score exponentially many points ?
« Reply #18 on: November 10, 2011, 01:56:48 pm »
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You don't need more cards in play to score an infinite amount of points (more precisely, to continue accumulating point indefinitely): village, goons, ambassador(return two copper), buy two copper. Repeat. Obviously you can ramp this combo up to gain more points per turn, but this simple combo can continue gaining VP forever.

Monument -> Buy nothing does the same; and simpler. ;)

*Edit* Ninja'd by 1 minute!

Good point. I wonder which could gain vp faster if both types of decks were ideally constructed. In other words, using tactician, KCs, wharfs, upgraded cities, etc to draw and play the maximum amount of each card. Goons/Ambassador seems stronger, but you spend more of your draw/actions to make it work than the simpler Monument. Probably Monument's reliability would win over Goons' ridiculous vp-gaining potential.

On second thought, that can't be right though.  Monument caps out at 30 vp per turn, which Goons could equal with a purchase of only three cards. So if you can find a way to guarantee playing all ten Goons AND returning just four copper (with two ambassadors), then Goons wins. That is made even more certain by the fact that the Goons engine can use all of its KCs on draw and actions, while Monument has to use part of its KC power on the vp gaining. So it should be no problem outpacing an ideal Monument engine with an ideal Goons/Ambassador engine.

Not that anyone ever wins in a match between infinite-vp engines... :)
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Jack Rudd

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Re: Can you score exponentially many points ?
« Reply #19 on: November 10, 2011, 02:04:03 pm »
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I reckon the KC-KC-Goons-Masquerade deck is going to be dominant here.
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Kirian

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Re: Can you score exponentially many points ?
« Reply #20 on: November 10, 2011, 02:06:47 pm »
+1

To extend DStu's idea into hyperdrive, let's do this.

First, to define a variable:  The number of cards in each stack is N.

Start of turn, you have 2x*Village, Watchtower, 2xCouncil Room.  On top of your deck, lots of Highways, and the other cards you need.
After playing Vil/Vil/CR/CR, you play 11 total Highways, and you have 11 cards in hand:

X, Watchtower, TR, TR, KC, KC, IW, IW, Workshop, CR, Goons.  The X is completely irrelevant.

TR|TR|KC|KC allows you to triple a total of 6 action cards.

Tripling IW/IW/WS allows you to gain 9 cards (the 9 engine cards in the above hand) and gives you 6 bonus actions, of which you'll need just one.
Watchtower allows you to put all 9 gains on the deck.
Triple CR draws all 9 and gives 3 Buys
Triple Goons gives 3 Buys.
Your sixth tripling goes unused; this engine won't support it.

You'll be able to do this N/2 - 1 times before the KC, TR, and IW piles run out.  The -1 is effectively irrelevant here, so I'll drop it in the following equations.  After all this is done, you have available:

More actions than you'll ever use.
3N Buys (6 * N/2).
N/2 copies of Goons in front of you.

Those 3N buys can empty any three Green piles you want other than Vineyard.  Your final deck will contain 4.5N (plus an insignificant constant) action cards and 3N Green cards.  Gardens and Silk Road will both be worth (3N/4) points each, so take your pick.  Since Duke increases as the square, that's an obvious one.  So:

Empty Duchy pile for 3N points.
Empty Duke pile for N^2 points.
Empty Gardens or SR pile for (3/4)N^2 points.
Each buy earns N/2 points from Goons, for a total of (3/2)N^2 points.

Total:  (13/4)N^2 + 3N points.  For N = 1E42, the second term isn't even within significance.  Your total score is within <0.0000001% of 3.25E84.

TR, KC, IW, CR, Duchy, Duke, and Gardens/SR are all empty:  game over, one turn after getting it all set up... though that part will take a significant amount of time.

Edit:  42 x 2 is 84.  Not 48.  Blah.
« Last Edit: November 11, 2011, 01:29:37 am by Kirian »
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tlloyd

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Re: Can you score exponentially many points ?
« Reply #21 on: November 10, 2011, 02:31:55 pm »
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I reckon the KC-KC-Goons-Masquerade deck is going to be dominant here.

I wasn't suggesting a head-to-head match, but rather a comparison of parallel solitaire games. In that case, goons/masq can't keep up with goons/amb, since the former will eventually run out of cards to buy, while the latter can accumulate vp indefinitely.
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gamesou

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Re: Can you score exponentially many points ?
« Reply #22 on: November 10, 2011, 06:01:53 pm »
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Anyway, since DStu has found a solution better that what I was thinking (he is able to score arbitrarily many points, while I score only exponentially many points), the puzzle is solved. Here is what I had in mind.

Let N be the number of cards in each pile.

Reduce your deck to 5 Highways. On each subsequent turns buy as many Markets as you can. This will take log(N) turns before you buy the whole pile (if N=10^42, log(N) is about 140). Next turn, buy all Worker's Village. Next turn, buy all Grand Markets but one. Next turn, but all Dukes, all Duchies and all Silk Roads (this can probably be fine-tuned).

Since this strategy is simpler to set up than DStu's, there should be a range (if N is not too large) where it is more efficient. But 10^42 is too large, I fear.

Here is a real-life implementation.

http://dominion.isotropic.org/gamelog/201111/10/game-20111110-063632-4cae1287.html

Without KC or TR of VP tokens, it looks that a reasonable score. But of course in usual dominion your opponent will interfere (simply by buying the cards you want) and this strategy is just nonsense.
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jimjam

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Re: Can you score exponentially many points ?
« Reply #23 on: November 10, 2011, 10:56:08 pm »
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Having either unbounded turns or unbounded piles makes the puzzle uninteresting. You have to track one and limit the other, or track both.
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DStu

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Re: Can you score exponentially many points ?
« Reply #24 on: November 11, 2011, 01:36:40 am »
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I'm not sure if yours is really faster to set up, or at least the range is really small. For mine you basically need a deck of 8 (or 11,7) (depending if you want to run out Provinces, Colonies or Duchy/Duke/SR) Highways, two KC, two IW, a CR. For the combo to start, you need to play all Highways before the other five cards, meaning one of the five must be the last card in you deck. The probability of the is 5/13 (5/16, 5/12), so this will happen in about 3 turns.

To get this deck, I need to buy the some more cards than you before the combo starts, but yours than continue with buying markets, WW, GM which also takes time. So if log(N)<=3, this might be faster, but i doubt that the range is very large.
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