91–100 of 146 posts
I always liked the prisoner box question, but I prefer the phrasing where the prisoners are assigned a number and the boxes are numbered. I feel like the "prisoners have to come up with a name to number mapping" step just gets in the way of the interesting part.
Another very counterintuitive (for me) problem: how do you do better that break even in the following game: 'A' chooses two distinct integers, writes them on slips of paper and holds one out in each hand in a fist. You choose a hand and reveal a number. You must then guess whether the other number is higher or lower than the revealed one, winning $1 if you guess right and losing $1 otherwise.
After writing this one out, this reminds me of the Monty Hall problem. In this case my guess is that you use a prior -- assume the two unknown numbers are A & B, and then assume a random integer yourself C. From there, if A (the first revealed number) is less than C, then that narrows the remaining cases giving you a 2/3 chance. If A is greater than C, that also narrows the remaining cases and gives you a 2/3 chance…
This is impossible; there's no uniform measure on the integers (as σ-additivity makes it impossible to bound such a measure).
Love in Kleptopia needs to be explained better. The problem can only be solved if you can afix two padlocks onto a box, and I was presuming the lock box had a single, normally shaped padlock eye, which would make such a thing impossible. I find this happens a lot with "thought" problems: I can't solve it (and can often prove that) because the rules of the problem are inadequately explained.
Or just attach a lock that needs a combo (ex: 3,7,15) instead of a key to open it, then tell her the combination after she gets it and the risk of theft is zero. The problem specifically says they are communicating using the internet, so why not? "Hey, I sent you a box. It's locked with a combo lock. Call me when you get it: I want to be talking to you when you see the surprise! I'll tell you the combo on the phone!"
The only way to avoid this possibility (of a man in the middle attack) is to be "introduced" by a third party you already trust. See e.g. https://en.wikipedia.org/wiki/Needham–Schroeder_protocol
In the absence of a trusted third party, you could be talking to anybody, through any number of lurking intermediaries. If you've got a trusted third party then the problem, as posed, is moot.
Another very counterintuitive (for me) problem: how do you do better that break even in the following game: 'A' chooses two distinct integers, writes them on slips of paper and holds one out in each hand in a fist. You choose a hand and reveal a number. You must then guess whether the other number is higher or lower than the revealed one, winning $1 if you guess right and losing $1 otherwise.
This problem frustrates me, I'm not quite convinced that it's well defined as written. My first instinct is to say "Based on your knowledge of Alice, assign a probability distribution over the pairs of integers she might pick. Then when one is revealed you should condition on that fact. Then just pick whichever of higher or lower is most likely." The problem setter will object that we have no way of assigning a proba…
Anyways, this is also highly unintuitive for me, so I’m far from certain that the solution below is correct; having said that, I’m unable to find a flaw in it. If you can, I’d be interested in hearing it.
SPOILER / SOLUTION:
Pick a number - say 0 for the sake of it. After Alice reveals the first number, assume that the other number is on the opposite side of 0 (i.e. if Alice reveals a negative number, assume the other number is positive and say “higher”, and vice versa). If your number is within the interval of Alice’s numbers, you win; otherwise, you still have a 50% chance of winning.
Another very counterintuitive (for me) problem: how do you do better that break even in the following game: 'A' chooses two distinct integers, writes them on slips of paper and holds one out in each hand in a fist. You choose a hand and reveal a number. You must then guess whether the other number is higher or lower than the revealed one, winning $1 if you guess right and losing $1 otherwise.
Assume that the range of integers is limited by the largest positive or negative number that can be written on the limited space of the paper. Whenever the largest positive or negative number comes up your answer will be certain. Turn a profit over a few billion years.
Another very counterintuitive (for me) problem: how do you do better that break even in the following game: 'A' chooses two distinct integers, writes them on slips of paper and holds one out in each hand in a fist. You choose a hand and reveal a number. You must then guess whether the other number is higher or lower than the revealed one, winning $1 if you guess right and losing $1 otherwise.
Is the range of integers limited? Does she choose them randomly? If the range isn't limited, hoe would we even define a normalized probability distribution?
Earlier quoted context omitted.
To get super pedantic, even if that was the case you could use a lock out tag out type device to still attach two locks. https://www.media-partners.com/upload/i20121017160441/img1.j...
Unfortunate that they don't show multiple locks in that image.
I’ve stopped trying to do puzzles. I’ve been misled by bad communication (intentional or otherwise) too many times.
Earlier quoted context omitted.
Ok now it is making sense. The same would be true if you went to three dots.... the third person would think, “wait, why aren’t the two blue dotted people killing themselves? There must be a third blue dot... wait, the third blue dot must be me” That makes sense. What about the other commenter, who said something like “Alice has a blue dot.” Wouldn’t that not lead to everyone’s death?
In your three dots logic, this would have a red dot person kill themselves if there are two blue dots: The two blue dotted people aren't killing themselves because they are still thinking "wait, why isn't he killing himself" about the other. I feel like some timing procedures need to be defined.
Earlier quoted context omitted.
This problem frustrates me, I'm not quite convinced that it's well defined as written. My first instinct is to say "Based on your knowledge of Alice, assign a probability distribution over the pairs of integers she might pick. Then when one is revealed you should condition on that fact. Then just pick whichever of higher or lower is most likely." The problem setter will object that we have no way of assigning a proba…
I agree that it’s not entirely strictly specified - in particular, there’s no random (uniform) distribution over integers. I guess you could get around this by specifying in in a way you did, or “parametrized” over some parameter, or say “for any distribution” (and mandate a single turn of the game)... Anyways, this is also highly unintuitive for me, so I’m far from certain that the solution below is correct; having…
Now personally I think this is silly. You know nothing about what Alice is going to do, so your own choice of number can't possibly matter. You might as well always pick zero rather than randomizing. The reason the orthodoxy disagree is because the problem they're answering is something like "Find a strategy such that if you knew which numbers Alice had picked, but hadn't yet chosen your own random numbers, then you would always anticipate a >50% chance of winning."
This resembles the definition of a 95% confidence interval: "A procedure for generating intervals from data, such that if you knew the true value of the parameter, but not the data, you would assign a 95% probability that the true value lies in the interval". Of course Bayesians turn it around and define a credence interval where you know the data but not the true value (which I think is better, because this actually is your state of knowledge).