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[lefaiison]

Personally, I hate treasure maps.  I think they're too luck based, but when playing an opponent who manages to get a turn 5 treasure map activation, I like to figuratively throw my keyboard out my window.

So first, I thought, is there a way to activate your Treasure Maps on Turn 3?  I could not come up with a solution.  So I figured it must be possible in turn 4.  Turns out that's true and there's a lot of ways to achieve this.  So how many ways is this possible?  And now the tough part.  Which of these solutions is most likely to happen (don't count probability of your opening draw, you can just set it to what you want it to be)?

My best solution is 4.8% of the time -- and surprisingly is probably going to be the first solution most people will think of.

[Jack Rudd]

Well, the obvious solution is

Open Talisman/Watchtower; turn 3 holding is Talisman-Watchtower-Copper-Copper-Copper. Buy Treasure Map, gaining another, and put both on the deck.

Probability of a 3/4 split is 5/6; probability of getting both your opening buys and 3 Coppers on Turn 3 is 7C3/12C5 = (7!/3!4!)/(12!/7!5!) = 7*6*5*4*5/12*11*10*9*8 = 35/792, for an overall probability of 175/4752.

[lefaiison]

Yeah, that was one of my solutions but it only happens 2.7% of the time.  Of course, unless my math is wrong. 

[Deadlock39]

Interesting, when I read this I figured the obvious solution was:

Open Chancellor/Treasure Map; Turn 3 play/activate Chancellor and buy a Treasure Map; Reshuffle both onto the turn 4 hand.

This would be lower chance than the Talisman/Watchtower, because you have to shuffle 2 cards out of 13 into your 4th hand instead of 2 out of 12 into your 3rd.

Edit:
I get a 7.6% chance for Jack's solution (assuming a 4/3 start. 6.3% assuming nothing), but my statistics are pretty weak.
Totally wrong... However, my overlooking that you must have 3 Copper in the 3rd hand and not my weak statistics was the culprit.
Jack's numbers look correct to me (3.68%).
My weak statistics aren't good enough to compute odds on my solution.

The odds of getting Chancellor + 2 Coppers in your third hand is not an easy one. The 4th turn shuffle odds on its own is 6.4% (I think)

[DG]

Start council room, pawn. Turn 3 play pawn for +1 coin, +1 action, draw up to seven copper with the council room. Buy two treasure maps and draw them on turn 4 with the three estates.


Start cellar, treasure map. Turn 3 play cellar and discard four cards, draw four copper, buy treasure map. Draw both maps on turn 4, or draw one map and use the cellar again to discard three cards and draw the second map.

[Axe Knight]

Open Treasure Map/Masquerade.  Draw the Masquerade on Turn 3 without the Treasure Map, and someone passes you a Watchtower.  Have enough Copper in your hand to buy another Treasure Map, putting it on top of your deck.  On Turn 4, draw the Treasure Map you bought last turn, along with the Treasure Map you bought in the opening.

The probability of this opening can only be calculated by a very horrifying sample of games...

[Thisisnotasmile]

Open Treasure Map/Masquerade.  Draw the Masquerade on Turn 3 without the Treasure Map, and someone passes you a Watchtower.  Have enough Copper in your hand to buy another Treasure Map, putting it on top of your deck.  On Turn 4, draw the Treasure Map you bought last turn, along with the Treasure Map you bought in the opening.

The probability of this opening can only be calculated by a very horrifying sample of games...

If we're allowing Masquerade shenanigans:

Open Treasure Map/Nothing in seat 2. Turn 3 Player 1 plays Masquerade and passes you Treasure Map (He opened Masquerade/Treasure Map). You now hold your own TM and his for a turn 3 activation.

[HiveMindEmulator]

Yeah, that was one of my solutions but it only happens 2.7% of the time.  Of course, unless my math is wrong. 

I think your math is wrong.
I compute probability of drawing your 2 opening purchases and 3 coppers on turn 3 is P(first purchase in top 5 cards)*P(second purchase in top 5 cards | first purchase in top 5 cards)*P(first 3 other cards are copper) = 5/12*4/11*(7/10*6/9*5/8) = 35/792 = 4.4%.
Or by counting (using nCr to denote the binomial coefficient "n choose r"):
number of successful turn 3 hands = 5C2*2*7C3 (choose locations of the 2 opening purchases in the top 5 cards, they can be in either order, choose the locations of the 3 estates in the remaining cards)
number of possible deck arrangements = 12C7*5C2*2 (choose the locations of the 7 copper out of 12, choose the locations of the 2 non-estates of the remaining 5, the 2 can be in either order)
So P(success) = 7C3/12C7 = 35/792 = 4.4%.
This 4.4% number actually shows up in the "basic opening probabilities" blog entry (http://dominionstrategy.com/2011/03/09/basic-opening-probabilities/) as the probability of drawing cccss on turn 3, and Jack Rudd got the same thing, so I'm pretty sure it's right. Of course, this is still not as good as your 4.8% solution, assuming your math is right for that solution.

[lefaiison]

Weird.  I thought I had replied in this thread, acknowledging that my math was wrong for the Talisman/Watchtower solution.

Anyway, There are a ton of solutions to getting a turn 4 TM activation.  Basically, you can either get something that cycles your deck and a Treasure Map, then buy another on Turn 3, and with enough cycling you can draw both on Turn 4.  Or you can use watch tower to align your 2nd Treasure map with your 1st.

Example:

Chancellor / Talisman / Treasure Map x2
Pawn / Council Room / Treasure Map x 2 (Probably lowest chance possible)
Copper / Council Room / Treasure Map x 2 (Slightly higher than above)
Warehouse / Treasure Map / Treasure Map
Treasure Map / WatchTower / Treasure Map
Cellar / Treasure Map / Treasure Map

However, Deadlock had it correct with what I thought was the most obvious solution.  Chancellor / Treasure Map / Treasure Map.  Turns out the odds of drawing Chancellor + 2 Coppers is fairly high (38%).  Then getting the two Treasure maps to align is (5/13 * 4/12) = 12.8%.  Multiply the two together and you get 4.86%, which was quite surprising to me.

Unless of course, my math is wrong