Print

Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

[irmo]

Sowelu: What are you smoking?

You're about to open an envelope with money in it, and are told the other envelope has either half that amount or double.

Define the money in your envelope as X. The amount of money in the other envelope is either X/2 or 2X. The average of X/2 and 2X is 1.5X.

So if you stick with the envelope you have, your expected value is X (by definition). If you switch, your expected (read: average) value is 1.5X. The variation between the X envelope and the other one ranges from X-(0.5X) and X+X. You can see what's going on here.

That assumes that the other envelope is equally likely to have double or half as much. I don't think you can do that when the envelopes are drawn from a finite distribution with a maximum.

In the coin-flipping version I don't think it's meaningful to talk about the expected value of "each flip", because part of the outcome of each flip is that all future flips have their stakes changed. But transform the whole problem into log space: your money is 2^X. Heads, X increases by 1; tails, X decreases by 1. Now the expected value of each flip is zero.

I think the right answer to "How long do you keep flipping?" is "How much money is enough for you?" Because there's no cost to flip, and you can't go to zero, you can just keep playing until you get a long enough run of heads to satisfy you.

[Sowelu]

So yes, switching will either halve or double your amount of money, but you gain or loes the same amount no matter what happens.

Not really. On an individual turn, you either gain L (where L is the amount you already have) or lose half of L.

Sorry, I should have quoted my earlier post:  L is defined as "the lower of the two envelopes", hence the L.  When you go from low to high, you gain L.  When you go from high to low, you lose L.

That assumes that the other envelope is equally likely to have double or half as much. I don't think you can do that when the envelopes are drawn from a finite distribution with a maximum.

Correct; if you open the first envelope and it's either below (minimum/2) or above (maximum/2), your strategy changes to "always swap" or "never swap" respectively.

And since this isn't a coin flipping thing, you only have two envelopes in your hands.  You can choose one or the other, but changing your choice doesn't make a difference except at the far ends of the range.  If you don't know what's actually in the envelopes, it never makes a difference.

[irmo]

That assumes that the other envelope is equally likely to have double or half as much. I don't think you can do that when the envelopes are drawn from a finite distribution with a maximum.

Correct; if you open the first envelope and it's either below (minimum/2) or above (maximum/2), your strategy changes to "always swap" or "never swap" respectively.

Though, thinking about that, I have to call shenanigans on the whole problem.

[Edit: No, I don't, I misunderstood it. The correct answer is that you look in the first envelope and switch only if it's less than $2.]

[Sowelu]

Though, thinking about that, I have to call shenanigans on the whole problem.

I pick an envelope randomly from a set of envelopes, and then you hand me another one drawn from the same set that contains either half as much or twice as much? To guarantee that the terms of the problem are satisfied, you have to peek at my envelope and go find one to match it. Suppose there are a hundred envelopes, one with $2 and all the rest with $1. Then I have a 99% chance of drawing one of the ones with $1, you have to go through them all and pick the one with $2 and offer me that, and I'll take it. So I get to use your ability to peek at envelopes to my advantage.

From the problem as originally stated:  "There's a huge pile of envelopes that contain various amounts of money, from one dollar up to 65 thousand (give or take).  Someone gives you two envelopes and tells you to pick one."  Regardless of the distribution of the rest of your pile, one of your envelopes contains some L from the start and the other contains 2L.

[ed boy]

I've decided to necro this thread with a fresh new puzzle, one of my favourites.

You are a strapping young lad who is going to marry one of the princesses of the land. There are three princesses: one always tells the truth, one always lies, and one sometimes tells the truth and sometimes lies.

You would be happy if you married the one that always tells the truth. Similarly, if you married the one that always lies, you can just treat everything she says as the opposite, and you would be perfectly happy with that. What you want to avoid is marrying the one that sometimes lies.

The king understands your predicament. He invites you to come to his throne room where his three daughters are waiting. Once there, you can ask a single princess (you don't know which one) a question of your choice before making your decision.

How do you proceed?

[iceball3]

I've decided to necro this thread with a fresh new puzzle, one of my favourites.

You are a strapping young lad who is going to marry one of the princesses of the land. There are three princesses: one always tells the truth, one always lies, and one sometimes tells the truth and sometimes lies.

You would be happy if you married the one that always tells the truth. Similarly, if you married the one that always lies, you can just treat everything she says as the opposite, and you would be perfectly happy with that. What you want to avoid is marrying the one that sometimes lies.

The king understands your predicament. He invites you to come to his throne room where his three daughters are waiting. Once there, you can ask a single princess (you don't know which one) a question of your choice before making your decision.

How do you proceed?

Ask the princess, "If I always lie, would I tell the truth if I said I was?". The always truth and always lie telling princess won't answer. The one who does answer isn't restricted to just lies and truths, because each are both sometimes.

[Leafsnail]

Spoiler: Answer (click to show/hide)

Point at the other two princesses.  "Which of those two girls lie more often?"

Marry the princess who the answering princess says lies more often.

Spoiler: Reasoning (click to show/hide)

If you're talking to the truth speaking princess, she'll point at the liar.  You marry her and live happily.

If you're talking to the liar, she'll point at the one who tells the truth.  You marry her and live happily.

If you're talking to the truth/liar, she'll point at either of them.  You marry either of them and live happily.

[ed boy]

Ask the princess, "If I always lie, would I tell the truth if I said I was?". The always truth and always lie telling princess won't answer. The one who does answer isn't restricted to just lies and truths, because each are both sometimes.

Let T be truthful, F be false, and S sometimes.

T-The initial if clause (if I always lie) does not hold, so nothing she says will be false. She can say anything in response to that.
F-The initial if clause (if I always lie) does hold, so she must answer the second part of the statment (would I tell the truth if I said I was) falsely. However, I don't quite understand what you mean by this - Is it (would I tell the truth if I said I was telling the truth) or (would I tell the truth if I said I was the one who always lies)?
S-Can say anything.

Spoiler: Answer (click to show/hide)

Point at the other two princesses.  "Which of those two girls lie more often?"

Marry the princess who the answering princess says lies more often.

Spoiler: Reasoning (click to show/hide)

If you're talking to the truth speaking princess, she'll point at the liar.  You marry her and live happily.

If you're talking to the liar, she'll point at the one who tells the truth.  You marry her and live happily.

If you're talking to the truth/liar, she'll point at either of them.  You marry either of them and live happily.

Good job.

[Leafsnail]

I've just looked up the riddle, and someone's pointed out that there's some further utility you can get out of the question:

Spoiler (click to show/hide)

Since you're not gonna end up marrying them, make sure you ask the question to the princess you find least attractive.

[Solifuge]

Since it's been answered, but no new riddle has been posed, here's a little one for y'all:

You've seen me before, I would wager.
Some take meals in my company.
I've made captives of crowds with my music,
And am known to make criminals free.

In my family, I am the least vicious,
But I'll work beside them, just the same.
Though their jobs often bring them to ruin and stab,
I will not. Can you guess at my name?

[Strife26]

Christian Rap?




My riddle is still open, from waaay back in the thread.

Spoiler (click to show/hide)

The answer's a number, but it's more than that. It has length. It has the original spark for half of my name. It's where cloaks and 16 tomes of magic spells came into my closet. Elegantly bound books. It's where I got more than Fire, Earth, Water, and Air. It's a game, it's a song, it's a movie . . .

What is it?

It's neither 13 nor 7. But one of them is a fourth of the way correct. Of course, the other is closer and further.

[malimbar04]

bah... Ether?:
number... (irrational numbers don't exist... like ether)
Length... no idea
spark for... no idea
where cloaks and magic spells fit into closet (the trans-dimensional space of magic is often called ether in stories)
Elegantly bound books... no idea
More than fire, earth, water, and air (the 5th classical element)
According to wikipedia it's the name of a game and a song, but I don't see a movie (yet).

It's not pi
I almost listed the numbers of each of the letters per word, or per period, etc. I don't know of any irrational number that fits the structure of your puzzle though.

and "half my name" - what name is that even refering to? Yours? itself? is this meaningless?

It's also not a color or any length I know of.

Meter? its' a number and a length. and is used to measure music or anything else arbitrarily.

I'm out of ideas for now, but I'm sure it's either a word puzzle or a structure of the question puzzle.

[G-Flex]

irrational numbers don't exist

Say what now?

[Solifuge]

You've seen me before, I would wager.
Some take meals in my company.
I've made captives of crowds with my music,
And am known to make criminals free.

In my family, I am the least vicious,
But I'll work beside them, just the same.
Though their jobs often bring them to ruin and stab,
I will not. Can you guess at my name?

Christian Rap?

I see where you we're going, but nope.

[Dr. D]

Some sort of songbird? (For Solifuge's riddle)