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1

Puzzles and Challenges / Cards with maximum buying value

« on: August 27, 2012, 06:39:53 am »

This one's just for fun.  And not so much a puzzle as a challenge/quiz of sorts.  A break from my usual style 

Challenge: 

You have some amount of $ to spend during your buy phase this turn.  You play a single card, and the amount of $ increases by X.  For a given card, define its maximum buying value to be the maximum X it can possibly achieve as the single card played.  From memory, give a list of the 10 Dominion cards with the largest maximum buying value, in order. 

Rules:

These are all very important.

• Don't look anything up!  Use your brain and scratch paper if you need it. 
• Don't count money obtained by playing a different card, or money obtained in indirect ways.  This is strictly about the amount of $ given directly by playing that card. (For example, under my definition, Haggler, Black Market, Merchant Ship, Cutpurse count for 2; Talisman, Venture, Counterfeit, Quarry, Ironworks, Bridge, Pawn count for 1; and Procession, King's Court, Golem, Horn of Plenty, Coppersmith, Counting House, Adventurer, University, Princess count for 0). 
  
• Must be a legal game.  Can be 6 player, involve any legal Kingdom.  Black Market can have all non-Kingdom cards, as the rules state (unlike Isotropic).  However, Treasure piles are the standard 60, 40, 30.
• Your list may not have ties, even though the actual answer may in fact have tied values.  This is because I couldn't think of a simple scoring system that doesn't encourage having lots of things being tied.  If you know something is a tie, you can just break it arbitrarily (it won't affect your score).

Scoring:

For each card which is indeed in the top 10, you get a number of points equal to 10 minus the minimal distance between your guessed ranking, and from the actual rank interval.  The maximum score is 100. 

E.g. Say you put card X at rank 5.  Then, for that card:

•   If its actual rank was 4, you would get 9 points.
•   If its actual rank was 8, you would get 7 points.
•   If it were actually tied for 7-10, then you would get 8 points.
•   If it were actually tied for 3-5, you would get 10 points. 
•   If its actual rank were 11, you would get 0!

Solutions:

Here is what I believe is the actual ranking, using cards up to Dark Ages: 

1-2. Pirate Ship, Band of Misfits (infinite)
3-5. Vault, Secret Chamber, Store Room (around $600)
6. Diadem (around $325)
7. Bank (around $280)
8. Philosopher's Stone (around $120)
9. Copper (around $100)
10. Forager ($23, I think)

Harsh!  You got 0 points for listing:

11. Trade Route ($14, I think)
   Black Market with Trade Route/Young Witch.  Tunnel bane, 9 other Kingdom VP cards.  Estate, Duchy, Province, Colony.
12. Salvager ($11)
13-14. Death Cart, Platinum: $5
Honorable mention: Poor House, Harvest, Tribute, Baron, Fool's Gold ($4)
Dishonorable mention: Poor House (around -$250?)

Explanations:
1-2. Pirate Ship, Band of Misfits
   Thanks to Rogue/Graverobber.  Pre-Dark Ages, could reach $263, or $234 in a 2 Player game.  See topic 2107.
3-5. Vault, Secret Chamber, Store Room (around $600)
   Size of largest possible deck is around 600.  6 player game.  Include a Black Market deck with all cards not in the Kingdom, including Young Witch and Tournament.  Let Rats, Hermit, Urchin, Marauder, a Prosperity card, and a Potion-cost card be piles, with VP Kingdom cards for the rest.  Ambassador/Masquerade shennanigans to get opponent's starting hands.  Native Village to draw, on the final turn. 
6. Diadem (around $325)
   Have a chain of Throne Rooms above a chain of Processions, and a parallel chain of King's Courts.  Tributes are played by King's Courts, Fortresses by Procession, and fill in rest of chain with arbitrary Villages.  Some Black Market trickery to get a Village bane, with Tournament (bringing Trusty Steed) and all remaining net-action cards in the Black Market deck (including Crossroads, Hermit bringing Madman, Dame Molly, etc).  Have Necropolis, of course.  Pre-Dark Ages was closer to $270, I think (narrowly beating Pirate Ship and Bank?  Not sure).  I believe including Graverobber/Rogue so that we can get back King's Courts/Tributes/Processions is not going to help, but haven't done the analysis seriously. 
7. Bank (around $280)
   Have a bunch of Kingdom Treasures including Philosopher's Stone, and a Black Market giving Spoils, Diadem, etc.  Use a Native Village from the Black Market deck to draw the hand. 
8. Philosopher's Stone (around $120)
   Again, the size of largest possible deck is around 600.
9. Copper ($99, maybe a bit more)
   Coppersmith!  Plus King's Court, Band of Misfits, Procession, Graverobber, Rogue, Throne Room... see http://forum.dominionstrategy.com/index.php?topic=4409.msg98564#msg98564.  It's possible you can do better than this, but I doubt it beats Philosopher's Stone.
10. Forager ($23, I think)
   Black Market with all the Kingdom Treasures, and something for Spoils, Potion, Platinum.

How'd you do?  Harder than you thought?  Anyone get 100?!

Edit:  Top 3 scores so far

• O (96)
• Schneau (85)
• qmech (84)

Average score so far:  67.111 (9 entries)

2

Puzzles and Challenges / Asymptotic Dominion: Asking all Nth turn puzzles at once

« on: August 19, 2012, 08:19:08 pm »

We often ask, "What is possible in N turns?"  But we restrict ourselves to small N, because it's no fun to reason the details out about larger N.  Even at N=5, while solutions are full of ideas, details become tedious to work out.  But we can make large N interesting too.  In fact, this puzzle is about N approaching infinity.

Let's consider the function f where f(N) is the maximum number of points it's possible to have at the end of turn N, in a game of Solitaire Dominion.  A few tweaks are required to make this interesting:

• We care about asymptotic lower bounds, rather than exact ones.  After all, it's no fun to reason about exact details for large N, and this makes different bounds more comparable.  (Although in theory we can still get incomparable solutions, in practice we won't.)  You may want to know a bit of math:  http://en.wikipedia.org/wiki/Big_O_notation.  But the intuition is just that solution A is better than solution B if it eventually gets more points than solution B, i.e. it outgrows it in the long run.
• It's no fun to play an unbounded number of turns with bounded piles.  (Otherwise, the best solution obviously uses some combination of Monument, Goons/Ambassador, and Bishop/Graverobber/Rogue, and the function is linear and annoying to work out details for.)  Thus let's say each Kingdom pile has infinitely many copies of each card.  (Unfortunately, the Black Market deck only makes sense being finite.)  I'll call this game Asymptotic Dominion. 
• Lastly, while we're breaking the rules of the game, why limit to 10 piles?  This is a constraint which often gets in the way of interesting ideas.  Let's instead consider the puzzle for entire sets of expansions.

Thus, in Asymptotic Dominion, what is the best lower bound you can prove for f(N), where the set of cards is:

• Base set
• Base set + Intrigue + Seaside
• Base set + Intrigue + Seaside + Alchemy, no Possession
• Base set + Intrigue + Seaside + Alchemy + Prosperity, no Possession
• All cards, no Possession or King's Court (Possession isn't in the Black Market deck either.)

Hints: 

• Even with base set only, it's possible to do far better than Omega(N^2), or even Omega(N^1000).  Some intuition, for all the non-mathematicians:  A "Golden deck" gets only Omega(N), since it gets a fixed number of points per turn.  A deck that Islands a Gardens every few turns gets Omega(N^2) points, since you'll have Omega(N) Gardens and Omega(N) cards. 
• Numbers 4 and 5 are the easiest puzzles, especially number 4.  An optimal solution for number 5 has essentially been discussed already elsewhere on the boards!
• Average case is equivalent to worst case, for all of these, except for Base set only.

3

Game Reports / Thief: The Perfect Storm

« on: August 15, 2012, 04:27:14 am »

A beautiful game from long ago, played by a friend of mine:

http://councilroom.com/game?game_id=game-20110526-190554-3168276f.html

This is the perfect storm that Thief has been waiting for.  Its moment to shine.  There are no ways to get money besides Treasure cards.  King's Court and Scrying Pool make it possible to play tons of Thief.  Quarry/Talisman allows for building up King's Courts and Thieves quickly, early on.  Warehouse and Lookout help to sift.  Remodel provides a way to eventually bootstrap the deck, when the engine is up and running, and when money has been sucked out of the game. 

My friend (empiric) has surveyed the Kingdom, and goes straight for mass King's Court and Thief.  Schiz ignores Thief, buying Treasure cards, and managing to nab a Colony and Province before the Thief engine becomes overwhelming.  Eventually, he joins the Thieving party, but is a bit late.  Having thought ahead, empiric has grabbed a Remodel in the mid-game.  He's also managed to grab 2 Scrying Pools before letting the Thieves finish doing their jobs.  After all the money out of the game, Schiz has proven to be too short-sighted.  Thief has taken the Treasures he spent so much economy buying, and he's left with nearly nothing to work with.  On the last turn, empiric Remodels into the remaining 7 Provinces.  27 turns.  Not bad, considering most efforts were directed towards draining each other's economies.

There's a lesson to be learned here.  Many would overlook Thief in this situation, as a cached reaction, despite its strength.  It would've been an opportunity to play a beautiful strategy, wasted.  Every card has its use cases, however niche!  Even Thief.  (Okay, fine.  Maybe not Scout...)

4

Puzzles and Challenges / My first challenge: Turn 2 Double Colony

« on: July 24, 2012, 12:24:56 am »

This challenge is to have a single player hit 2 Platinum on turn 2, in a normal Dominion game.  (Yes, turn two.  Noble Brigand is a must.)  We can assume perfect shuffles, and helpful opponents (up to 6 players in the game, as the rules state). 

Hard Mode: 2 Colonies

I don't know that hard mode is actually possible, but I suspect it is.  I have a solution for easy mode, which is due to my friend (bitwise) and me. 

Update:  Hard mode is so very possible...