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[pacovf]

If any of you lack the capacity to solve problems involving infinitely countable hats (I know I do), here's one of my favourite puzzles (this one is by Raymond Smullyan; I had to retranslate the version I have, so sorry if the English is wonky):

In 1918, the day the First World War armistice was signed, three married couples celebrated the occasion dining together. Each husband is the brother of one wife, and each wife is the sister of one husband; that is, there are three brother-sister pairs in the group. We know the following:

-Helen is exactly twenty-six weeks older than her husband, who was born in august.
-The sister of Mr. White, who is married to the brother-in-law of Helen’s brother, got married to him on her birthday, which is in January.
-Marguerite White isn’t as tall as William Black.
-Arthur’s sister is taller than Beatrice.
-John is fifty years old.

What is Mrs. Brown first name?

I don't know if anyone will try to solve this one or not, but just in case, here's a hint that normally should get you to the answer:
The date is relevant.

[theory]

John's girlfriend lives at one end of town. His mother lives at the other. John gets off work at a random time between 5 p.m. and 6 p.m. He always visits either his girlfriend or his mother after work.  Mindful that he hasn't been spending enough time with his mother, he comes up with a new policy: he takes the first train that he sees when he leaves work.  Since both trains run exactly every ten minutes, he figures that this should be a good random way to visit both of them equally. 

A month later, Mom complains. She says that John has visited his girlfriend 27 out of 30 times this month.  John says it is just by chance, and that he has been following his policy of randomly getting on the first train he sees. 

Provide a logical explanation for this discrepancy, assuming that everyone is telling the truth and that the sample size is large enough. 

[Drab Emordnilap]

Clearly, the Mom train comes one minute before the Girlfriend train.

[Watno]

You mean after.

[Drab Emordnilap]

Yes, that's what I actually mean.

[pacovf]

Timetable: N, 5:09 | S, 5:10 | N, 5:19 | S, 5:20| N, 5:29 | S, 5:30 | N, 5:39 | S, 5:40 | N, 5:49 | S, 5:50 | N, 5:59 | S, 6:00 ; or something equivalent.

Gaah, ninja'd three times.

[Watno]

Though actually, that makes no sense, since there is no information about any corellation between the train he takes and who he visits.

[eHalcyon]

John's girlfriend lives at one end of town. His mother lives at the other. John gets off work at a random time between 5 p.m. and 6 p.m. He always visits either his girlfriend or his mother after work.  Mindful that he hasn't been spending enough time with his mother, he comes up with a new policy: he takes the first train that he sees when he leaves work.  Since both trains run exactly every ten minutes, he figures that this should be a good random way to visit both of them equally. 

A month later, Mom complains. She says that John has visited his girlfriend 27 out of 30 times this month.  John says it is just by chance, and that he has been following his policy of randomly getting on the first train he sees. 

Provide a logical explanation for this discrepancy, assuming that everyone is telling the truth and that the sample size is large enough.

WLOG, suppose that John's girlfriend lives in the east and his mom lives in the west.
This discrepancy may be explained if the train runs back and forth on a straight line and John works in the east as well:

Mom -----------------------John--Girlfriend

When John arrives at the station, it is more likely for the next train to be heading east.  Even though they both run every 10 minutes, it does not mean the trains arrive 5 minutes apart.  From John's station, trains will arrive from the west, run east for another station or two, then turn around and come back.  The window of time between arrivals is different.

My explanation here is probably not very clear, but you can look up the Elevator Paradox, which is the same idea going vertical.

[eHalcyon]

I suppose the 1 minute explanation is better because it matches up with the 27/30... I still like my explanation more.

[Watno]

I'm surprised I didn't know the english expression for WLOG before, I just knew the German OBDA

[Tables]

Not so much a logic problem, but a fun problem anyway:

Using each of the numbers 1, 3, 4 and 6, each exactly once, as well as any number of +, -. x and ÷ as well as brackets, make the number 24.

I think there's only one solution, and it took me at least 40 minutes to find

[eHalcyon]

Not so much a logic problem, but a fun problem anyway:

Using each of the numbers 1, 3, 4 and 6, each exactly once, as well as any number of +, -. x and ÷ as well as brackets, make the number 24.

I think there's only one solution, and it took me at least 40 minutes to find

6/(1-3/4)

Now try making 24 with just four zeros! You may use more complicated functions like trig functions, exponents, logarithms. Anything on a standard scientific calculator, excluding shortcut functions with other numbers like x^2.

[bozzball]

Not so much a logic problem, but a fun problem anyway:

Using each of the numbers 1, 3, 4 and 6, each exactly once, as well as any number of +, -. x and ÷ as well as brackets, make the number 24.

I think there's only one solution, and it took me at least 40 minutes to find

6/(1-3/4)

Now try making 24 with just four zeros! You may use more complicated functions like trig functions, exponents, logarithms. Anything on a standard scientific calculator, excluding shortcut functions with other numbers like x^2.

((0!+0!)^(0!+0!))!

[Jimmmmm]

Not so much a logic problem, but a fun problem anyway:

Using each of the numbers 1, 3, 4 and 6, each exactly once, as well as any number of +, -. x and ÷ as well as brackets, make the number 24.

I think there's only one solution, and it took me at least 40 minutes to find

6/(1-3/4)

Now try making 24 with just four zeros! You may use more complicated functions like trig functions, exponents, logarithms. Anything on a standard scientific calculator, excluding shortcut functions with other numbers like x^2.

((0!+0!)^(0!+0!))!

Or even just (0!+0!+0!+0!)!

[Ozle]

What?
Explain please

[Jimmmmm]

0! = 1. So (0!+0!+0!+0!)! = 4! = 4*3*2*1 = 24.

[Ozle]

0! = 1. So (0!+0!+0!+0!)! = 4! = 4*3*2*1 = 24.

Yeah, i assumed that meant 1, but why?

[theory]

http://en.wikipedia.org/wiki/Factorial#Definition

http://www.zero-factorial.com/whatis.html

It depends on how you define 0!, and it turns out that the most consistent, most logical, and most convenient (if not most immediately intuitive) definition of 0! is 1

[Grujah]

0! = 1. So (0!+0!+0!+0!)! = 4! = 4*3*2*1 = 24.

Yeah, i assumed that meant 1, but why?

Well, it just is.

I assume, cause for each n, you can n! = (n-1)! * n.  i.e. 5! = 4! * 5. So, to maintain 1! = 0! * 1, 0! needs to be 1.

[Drab Emordnilap]

Basically, x!, or "X Factorial", asks "Given x items, how many unique orders can they have, if put in a line?"

So, 2! = 2, AB or BA.

3! = 6, ABC ACB BAC BCA CAB CBA

1! is trivial, A.

How many arrangements can 0 objects have? One unique arrangement, in fact: ∅

Also,

4!/3! = 4
3!/2! = 3
2!/1! = 2

And we all know how mathematicians love patterns.

1!/0! = 1

Therefore, 1! = 0!

[Ozle]

Just because its true for one set of numbers, doesn't make it true for 0!

This seems a major assumption without any proof!
(Will read the wiki later though...)

This sounds like one of those things everybody agrees on and makes it a big joke on Ozle again!
Yeah, lets all pretend 0! = 1 to Ozle, hahaha
Its like that time they told me that the end of year prom was cancelled, or they didn't let people with glasses onto Busses for health and safety reasons. Or that time you could get shoes that fasten without laces!

Yeah, lets all laugh at Ozle!

[Drab Emordnilap]

Just because its true for one set of numbers, doesn't make it true for 0!

1337!/1336! = 1337

58008!/58007! = 58008

3720!/3719! = 3720

[Watno]

If you don't like it, do (cos(0)+cos(0)+cos(0)+cos(0))!
Anyway, how would you define a faculty?

[Jimmmmm]

0! = 1. So (0!+0!+0!+0!)! = 4! = 4*3*2*1 = 24.

Yeah, i assumed that meant 1, but why?

Convention. It's the most sensible and useful definition, so it's the one we use.

[DStu]

Not so much a logic problem, but a fun problem anyway:

Using each of the numbers 1, 3, 4 and 6, each exactly once, as well as any number of +, -. x and ÷ as well as brackets, make the number 24.

I think there's only one solution, and it took me at least 40 minutes to find

6/(1-3/4)

(14-6)x3