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[eHalcyon]

Do I have to play the choice card immediately?  If not:

My two cards were Minion and Duplicate.  I was going to play Minion for coins followed by Duplicate, just to get it onto the Tavern mat.  But you played Margrave, allowing me to draw Necropolis (or some other village).  I am thus able to play Necropolis and Duplicate before playing Minion to redraw.

Edit: My two cards are Minion and Vault.  I really want to buy a $5 card.  I plan on discarding with Minion because I expect Vault will only draw junk (and only reach $4 with Minion).  You play Torturer, giving me a Curse in hand.  Now I can play Minion for +$2 followed by Vault to reach $5.

Note, Vault actually does have a difference with Throne Room, but only with opponents.  You could have a similar scenario but with Secret Chamber and only wanting to hit $3, or maybe have a Merchant Ship in play or something.

[Asper]

Do I have to play the choice card immediately?  If not:

My two cards were Minion and Duplicate.  I was going to play Minion for coins followed by Duplicate, just to get it onto the Tavern mat.  But you played Margrave, allowing me to draw Necropolis (or some other village).  I am thus able to play Necropolis and Duplicate before playing Minion to redraw.

Edit: My two cards are Minion and Vault.  I really want to buy a $5 card.  I plan on discarding with Minion because I expect Vault will only draw junk (and only reach $4 with Minion).  You play Torturer, giving me a Curse in hand.  Now I can play Minion for +$2 followed by Vault to reach $5.

Note, Vault actually does have a difference with Throne Room, but only with opponents.  You could have a similar scenario but with Secret Chamber and only wanting to hit $3, or maybe have a Merchant Ship in play or something.

My solution only plays the choice card, but Duplicate/Minion is okay. Vault draws two cards, so it stacks with Throne Room. Or am i misunderstanding something?

[eHalcyon]

Do I have to play the choice card immediately?  If not:

My two cards were Minion and Duplicate.  I was going to play Minion for coins followed by Duplicate, just to get it onto the Tavern mat.  But you played Margrave, allowing me to draw Necropolis (or some other village).  I am thus able to play Necropolis and Duplicate before playing Minion to redraw.

Edit: My two cards are Minion and Vault.  I really want to buy a $5 card.  I plan on discarding with Minion because I expect Vault will only draw junk (and only reach $4 with Minion).  You play Torturer, giving me a Curse in hand.  Now I can play Minion for +$2 followed by Vault to reach $5.

Note, Vault actually does have a difference with Throne Room, but only with opponents.  You could have a similar scenario but with Secret Chamber and only wanting to hit $3, or maybe have a Merchant Ship in play or something.

My solution only plays the choice card, but Duplicate/Minion is okay. Vault draws two cards, so it stacks with Throne Room. Or am i misunderstanding something?

Errrrr, no.  That was a brain fart.

Secret Chamber though.  It works there.

[Asper]

Could be Torturer, Hamlet, Library... Not sure if discarding two cards instead is what was meant by different choice.

The choice card is your choice when you play it. Hamlet and Library are "may", so you can argue they are choice cards, and of course Workshop and Chapel give implicit choices, but i'm thinking of cards where the choices are spelled out on the card, with a "choose" on it, like Stuart. Narrows it down a bit.

[eHalcyon]

My two cards are Count and Secret Chamber (or Watchtower, or Library, or whatever).  I was going to use Count to discard SC, but you played Margrave and I drew a card worth saving (maybe with the help of Secret Chamber) so I topdeck with Count instead.

[Asper]

My two cards are Count and Secret Chamber (or Watchtower, or Library, or whatever).  I was going to use Count to discard SC, but you played Margrave and I drew a card worth saving (maybe with the help of Secret Chamber) so I topdeck with Count instead.

Count/SC makes sense, and even plausibly uses the non-scaling card. Nice, if not what i thought about :-)

[Asper]

Oh, and now that a legitimate solution was posted, here's mine:

Your hand is Squire and Counting House. You plan to play Squire for Actions, to play Counting House, get to $6 and buy a Gold. Your opponent plays Haunted Woods. You change your plans, choose two buys instead and buy three Coppers. You topdeck your Counting House, so next turn you will certainly be able to buy a Province, maybe even a Colony. Of course this assumes your deck has at least 4 cards in it.

[Dingan]

I have z Duchies and Dukes.  That is, I have x Duchies and y Dukes, where x + y = z.  For a constant z, what should x and y be to maximize my VP?  I.e. what is y in terms of x (or x in terms of y)?  Explain why (I like math).

[ephesos]

Time for some Lagrange multipliers!
Optimization function: VP(x,y)=x*y+3*x
Constraint: z(x,y)=x+y=constant
dz/dy=dz/dx=1
dVP/dy=x,dVP/dx=3+y
These vectors must be in the same direction, with a Lagrange multiplier -L applied:
(dz/dx,dz/dy)=L(dVP/dx,dVP/dy)=(L dVP/dx,L dVP/dy)
L dVP/dx=dz/dx, L dVP/dy=dz/dy
1=Lx, L=1/x
1=(3+y)/x, x=3+y

[liopoil]

Surely you don't need that fancy stuff to see intuitively that x - 3 = y, and that if this is not possible for a given z, then it is possible for z-1 and that the zth duchy/duke can be either one. If we assume that if we maximize the VP for each z < z_actual in sequence we get the right answer (not quite sure how to proove that, induction or something), then we see that after z₃ we simply alternate buying duchies and dukes. (also provable by induction)

[Deadlock39]

The number of points from Duchies is 3x, and the number of points from Dukes is xy.

Points = 3x + xy. 
    Substitute y = z - x
Points = 3x + x(z - x)
Points = 3x + xz - x^2
Points = -x^2 + (z + 3)x

By inspection, this is a parabola with a single local max. Derive and solve for 0 to find local max (Knowing how parabolas work is left as an exercise for the reader.)

0 = -2x + (z + 3)
2x = (z + 3)
*x = (z + 3) / 2
    Second solution by substituting x = z - y
z - y = (z + 3) / 2
-y = (z + 3) / 2 - z
y = z - (z + 3) / 2
y = 2z / 2 - (z + 3) / 2
y = (2z - (z + 3)) / 2
*y = (z - 3) / 2

Duchies = (z + 3) / 2
Dukes = (z - 3) / 2

Because Dukes and Duchies cannot be non-integer values even values of z do not properly solve the equation.  Because we know parabolas are symmetrical (still left as an exercise for the reader), we know in this case that substituting Duchies + 0.5 or Duchies - 0.5 will result in the same maximum number of points.

For z < 3, this solution produces an invalid negative number of Dukes. In this case Dukes must be assigned the minimum possible value of 0.  Substitution yields Duchies = z for this case.

(This notation is probably blasphemous, but... meh)

For z < 3:
Duchies = z
Dukes = 0

For z >= 3
Duchies = (z + 3) / 2 } if z odd
Dukes   = (z - 3) / 2 } if z odd
Duchies = (z + 3) / 2 +/- 0.5 } if z even
Dukes   = (z - 3) / 2 -/+ 0.5 } if z even

[Dingan]

Time for some Lagrange multipliers!
Optimization function: VP(x,y)=x*y+3*x
Constraint: z(x,y)=x+y=constant
dz/dy=dz/dx=1
dVP/dy=x,dVP/dx=3+y
These vectors must be in the same direction, with a Lagrange multiplier -L applied:
(dz/dx,dz/dy)=L(dVP/dx,dVP/dy)=(L dVP/dx,L dVP/dy)
L dVP/dx=dz/dx, L dVP/dy=dz/dy
1=Lx, L=1/x
1=(3+y)/x, x=3+y

This is my favorite answer, which I actually like to put in non-calculus terms:

VP(x,y) = xy + 3x = (x)(y+3)
Let (x) and (y+3) be the sides of a rectangle, so that VP is the area of the rectangle.  You want to maximize the area of the rectangle with a constant perimeter (perimeter is constant because perimeter = 2(x) + 2(y+3) = 2z+6 = constant).  Everyone knows that with a given perimeter, the largest rectangle you can make is a square.  So x = y + 3.

[liopoil]

Ooo, that is good. I think all three of our solutions are pretty much getting at that same idea.

[werothegreat]

23456
23456
2345
345678
23456

What's the next set of numbers, and what is the pattern?

[AJD]

23456
23456
2345
345678
23456

What's the next set of numbers, and what is the pattern?

123456

[Witherweaver]

23456
23456
2345
345678
23456

What's the next set of numbers, and what is the pattern?

123456

That's the kind of thing an idiot would have on his luggage!

[Elanchana]

23456
23456
2345
345678
23456

What's the next set of numbers, and what is the pattern?

23456
(0)123456
2345

...right?

It's the costs of all kingdom supply cards in each set by release date, yeah? Except Alchemy's not in there because the prices are all weird.

[Witherweaver]

Time for some Lagrange multipliers!
Optimization function: VP(x,y)=x*y+3*x
Constraint: z(x,y)=x+y=constant
dz/dy=dz/dx=1
dVP/dy=x,dVP/dx=3+y
These vectors must be in the same direction, with a Lagrange multiplier -L applied:
(dz/dx,dz/dy)=L(dVP/dx,dVP/dy)=(L dVP/dx,L dVP/dy)
L dVP/dx=dz/dx, L dVP/dy=dz/dy
1=Lx, L=1/x
1=(3+y)/x, x=3+y

This is my favorite answer, which I actually like to put in non-calculus terms:

VP(x,y) = xy + 3x = (x)(y+3)
Let (x) and (y+3) be the sides of a rectangle, so that VP is the area of the rectangle.  You want to maximize the area of the rectangle with a constant perimeter (perimeter is constant because perimeter = 2(x) + 2(y+3) = 2z+6 = constant).  Everyone knows that with a given perimeter, the largest rectangle you can make is a square.  So x = y + 3.

I mean, you could just as easily say   "Everyone knows that f(x,y) is optimized subject to the constraint g(x,y) when Df = c*Dg."

[werothegreat]

23456
23456
2345
345678
23456

What's the next set of numbers, and what is the pattern?

23456
(0)123456
2345

...right?

It's the costs of all kingdom supply cards in each set by release date, yeah? Except Alchemy's not in there because the prices are all weird.

Skipping Alchemy was supposed to be a clue that I was skipping small sets, so the next set would be Dark Ages.

[Elanchana]

23456
23456
2345
345678
23456

What's the next set of numbers, and what is the pattern?

23456
(0)123456
2345

...right?

It's the costs of all kingdom supply cards in each set by release date, yeah? Except Alchemy's not in there because the prices are all weird.

Skipping Alchemy was supposed to be a clue that I was skipping small sets, so the next set would be Dark Ages.

Ohhhhhhhh. I thought it was because writing P2P3P4P6P would have been a dead giveaway but your reason is better.

Also I'm kind of amused that my phone thinks (0)123456 is a phone number.

[ephesos]

I mean, you could just as easily say   "Everyone knows that f(x,y) is optimized subject to the constraint g(x,y) when Df = c*Dg."

Well, it would be easiest to just say "Everyone knows x=y+3." But that would be boring.

You have to start somewhere though, otherwise you can't get anywhere. Whether you start at knowing squares are optimizers or parabolas are maximized at the vertex or functions are optimized with respect to a constraint if their gradient vectors are proportional just depends on how much of the underlying mathematics you want to show.

[Witherweaver]

I mean, you could just as easily say   "Everyone knows that f(x,y) is optimized subject to the constraint g(x,y) when Df = c*Dg."

Well, it would be easiest to just say "Everyone knows x=y+3." But that would be boring.

No, because that's too out of context.  But once you define x and y, sure.

You have to start somewhere though, otherwise you can't get anywhere. Whether you start at knowing squares are optimizers or parabolas are maximized at the vertex or functions are optimized with respect to a constraint if their gradient vectors are proportional just depends on how much of the underlying mathematics you want to show.

Yeah, that was my point. 

[Asper]

Band of Misfits consist of heavy drinkers. How can you tell?

[ConMan]

Band of Misfits consist of heavy drinkers. How can you tell?

Because they can never remember what they did the last time they went out?

[Asper]

Band of Misfits consist of heavy drinkers. How can you tell?

Because they can never remember what they did the last time they went out?

This is a very good answer and i wish it was mine
Alas, it's not.

Edit: In case somebody wants a hint: This doesn't work in every kingdom.