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38

The answer is indeed...             ...because the question is equivalent to...   Calculations:


35

This surprisingly beguiling puzzle may also be solved with a surprisingly unsophisticated approach. Symmetry, by itself, predicts the average length of evens-only sequences ending with 6 to be... Start with T  many random throws: 2153664315121226553111444142566363625461525 . . 3644464461 Sift them into 4 groups that, due ...


32

Another way to think about it


25

The biggest problem with the prisoner's proof is that his model is incomplete. One piece of that is that it merges two key variables - days on which the execution can happen, and days on which the prisoner believes it possible to happen. It's easy to see why these two were merged - because of the requirement of surprise, they are strongly correlated. However,...


20

This is indeed possible if we define information content as the least number of bits necessary to describe the data. Consider the following example (in human readable language): Book 1: Jim knows all animals whose name starts with a letter between A and L. Book 2: Jim knows all animals whose name starts with a letter between M and Z. Then the information ...


19

I was at a dinner party the other night and I spoke to Mr Smith. He mentioned having two kids, both with unisex names: Sam and Alex. I remember that he told a story about "my son", but I don't recall which child he was talking about, and I don't know if he has a daughter or not. What are the odds that he has two sons? This avoids selection entirely, and ...


17

The books in the library contains all binary strings of length 1 million, one per book, sorted in lexicographic order. An individual book takes a million bits to specify, but this description of the whole library is much shorter.


17

The classic answer is "The surgeon is his mother". Another possible answers: It is another father of him. Gays are real.:) It is step-father. No one would care to chose precise words in such situation.


17

There is no paradox. The teacher will get paid, one way or another. The key to understanding the situation is realizing there are multiple slightly different scenarios that are all being described as being identical: The student is obligated to pay the teacher. The student will be obligated to pay the teacher. The conditions of the contract have been met. ...


15

Let $X_n$ be the event that the dice takes $n$ rolls to get the first 6, given all the rolls are even. Let $A_n$ be the event that it takes $n$ rolls to get the first 6, and let $B$ be the event that all rolls up to the first 6 are even. $P(X_n)=P(A_n|B)=\dfrac{P(B|A_n)P(A_n)}{P(B)}$ (using Bayes' theorem) Now: $P(A_n)=\dfrac{1}{6}\cdot\left(\dfrac{5}{6}\...


14

It seems rather obvious to me. Am I missing something? Edit to add this: I see now where there could be an incorrect way to reason about it that some people (like the cashier maybe) might do. It would have been nice if that were more clearly stated in the question. The cashier may have thought as follows: Putting together a batch of 4 at \$0.30 and 6 at \$0....


14

Original solution by YowE3K   (who later turned it into this community wiki) This isn't an answer to the exact question, but the following link is to an image that I thought was worth looking at anyway: And @justhalf found another image which looks even more like the one in the question, except rotated 90 degrees: I'm thinking that the actual ...


12

The negation of "All x are y" is "There is at least one x which is not y". So, Epimenides is a liar. Therefore his statement "All Cretans are liars" is false. This means that not all Cretans are liars. This means that at least one Cretan tells the truth. He can still be a liar, there just has to be at least one Cretan who's not a liar. Now, if Epimenides ...


11

Basically the mirror line is a physical equivalent of an ideal wall, which can react to any force you apply to it with the same, but opposite force. A physical object can not cross a wall, especially an ideal wall. So my answer is no.


11

The twins were travelling eastwards across the International Date Line during their birth. The elder twin brother was born first, on March 1st. After they crossed the International Date Line, the younger brother was born, on February 28th. When they celebrate their birthdays on a leap year, they're 2 days apart!


11

Case 1. Case 2. Bonus: [EDIT] To answer the question about who's right and wrong...


10

I think


9

Sorry if this is not in proper mathematical terms, I am not a mathematician...


9

Legally,


9

I believe the answer is This is computational calculation, so it is not statistical answer. It is for the people who try to find it probabilistically. Here is the probabilistic solution: First of all, we know that the probability of getting 6 on the first roll is $\frac{1}{6}$, then getting 6 after an even roll is $\frac{2}{6}\frac{1}{6}$, and so on as ...


8

There's room even for one more day of difference, that is, the elder brother can celebrate his birthday 3 days after the younger brother does. leoll2 says in his answer that they travel across the International Date Line to jump back to the previous day in the calendar. You don't have to travel to that part of the globe, you can do this at any time zone ...


8

It is rational for both parties to press the button, causing them to win $500$ on average. Here is what is wrong with the reasoning in the Paradox section. Once Alice is called in, she reasons that the strategy will lose them money. That is ok, since when Alice is not called in, she reasons that the strategy will win them money (when she is not called in, ...


8

This could work: Alternatively (with the same reasoning):


7

Create a strong magnetic field at the mirror surface. North is up, south is down. Fire an electron beam at the mirror through the field. Electrons from either side will be deflected anti clockwise (as seen from above) and will miss each other, passing through the mirror.


7

The simple (and I believe the only) answer to both your questions is "This is a paradox, so logic does not have predictability power here". Martin Gardner described this paradox in detail. Unfortunately, I can't find the English version online, but if you know Russian, you can find it here (The English Google translation is here). The English version should ...


7

(1) No statement $n$ with $2\le n\le N$ in the second part can be true, as it would imply its own falseness. Every statement in the second part of the list is false. (2) As statement $n$ with $2\le n\le N$ in the second part is false, we conclude that not all statements with a number divisible by $n$ are false. Equivalently, there exists some statement ...


7

You should be deducting the Bellhop's £20 from the £270, not adding it. Edited with more details: The question is deliberately misleading you into thinking there's a paradox, and some money has gone missing. The total amount paid initially was £300, £100 each. The total amount paid after the partial refund was £270, £90 each. The sly bellhop kept the ...


7

Explanation:


7



7

This doesn't seem to be that difficult.


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