The Maths Thread

[8ankitj]

Tend to think Marcuss is right.

It's easy to extend n=2 to n=3. If there are 3 people with blue eyes, each one of them expects the other 2 to commit suicide on 2nd day if there are exactly 2 blue-eyed men (as Marcuss explained). When no one commits suicide on 2nd day, each one of them knows there are more than 2 such men and thus each one commit suicide.

In case of n=4, each of them expects other 3 to commit suicide on 3rd day and so on...

Perfectly logical but I am intrigued by what did the outsider really do to trigger this chain of thought.

[Chewie]

I don't understand... Why does it take this many days?

Or am I just stupid

[Neil Pickup]

Originally Posted by Marcuss

It does work. Make it 3 people.
Me, B, C - all can see 2 other people with blue eyes. Everyone else can see 3 obviously.
On the first day nobody can guarantee anybody's eye colour. Using the logic explained in my previous posts, on the second day I expect B and C to commit suicide, B expects Me and C and C expects Me and B.
When B and C don't the only explanation is because each of them can see another person with blue eyes - Me.
Don't forget we're all equally logical and are aware that each of us is logical etc. So we're all undergoing the same though process so we all realise we have blue eyes on the 3rd day and kill ourselves.

That's because everyone has blue eyes. I don't dispute that.

The island we are talking about, though, has brown-eyed people in addition to the blue-eyed. Neither of your small cases deal with this.

[Marcuss]

Yes they do, they just don't need to be mentioned. They're not impacted upon in anyway.
When I say "Make it 3 people" change it to "3 people out of 1000". Nothing changes.

[Neil Pickup]

Yeah, get it. FMD.

[vcs]

There's another possible way of looking at it.. like Ankit says, the outsider does not add any new information. If there was only one bloke with blue eyes, then yes, when the outsider says it's nice to see another person with blue eyes, he'd go and commit suicide immediately. But here, each of the 100 guys knows there's atleast 99 others with blue eyes, and he knows that each of those 99 know of atleast 98 others with blue eyes and so on. So you could argue that no one would do anything and everything would carry on as normal. I'm starting to lean towards this argument now.

[Neil Pickup]

See, that was where I started, but Marcuss has got me to the other end. It is a fair question, though, why does that "some people here have blue eyes" start off the chain reaction, when everyone knows that some people have blue eyes anyway?

It's certainly one I'll want an answer for before I try teaching this one!

I think I see it, as without the trigger there is no expectation for anyone else to commit suicide, but that information creates the expectation that did not otherwise exist.

[vcs]

Anyway, this is the link where I copied the problem statement from, and you can see it's been done to death in the comments there.

[Marcuss]

Originally Posted by vcs

There's another possible way of looking at it.. like Ankit says, the outsider does not add any new information. If there was only one bloke with blue eyes, then yes, when the outsider says it's nice to see another person with blue eyes, he'd go and commit suicide immediately. But here, each of the 100 guys knows there's atleast 99 others with blue eyes, and he knows that each of those 99 know of atleast 98 others with blue eyes and so on. So you could argue that no one would do anything and everything would carry on as normal. I'm starting to lean towards this argument now.

There is no "and so on". Everybody can see at least 99 people with blue eyes, so if all those are still living after the 99th day you are the 100th.

All the people with brown eyes are going through the same though process but obviously they don't end up killing themselves because the 100 blue eyed people all die on the 100th day.

[vcs]

Yes, but when does "Day Zero" start? Is it the day the foreigner makes the parting speech? Do they all assemble at the hanging place everyday after he leaves, file out dutifully for 99 of those 100 days, and then go and hang themselves on Day 100?

[Marcuss]

Originally Posted by Neil Pickup

See, that was where I started, but Marcuss has got me to the other end. It is a fair question, though, why does that "some people here have blue eyes" start off the chain reaction, when everyone knows that some people have blue eyes anyway?

It's certainly one I'll want an answer for before I try teaching this one!

I think I see it, as without the trigger there is no expectation for anyone else to commit suicide, but that information creates the expectation that did not otherwise exist.

Yeah, they're not allowed to talk about eye colour so nobody thinks about it. The traveller's question sparks off the same though process in all of the islanders.

Originally Posted by vcs

Yes, but when does "Day Zero" start? Is it the day the foreigner makes the parting speech? Do they all assemble at the hanging place everyday after he leaves, file out dutifully for 99 of those 100 days, and then go and hang themselves on Day 100?

Yeah, pretty much.

[8ankitj]

Originally Posted by Neil Pickup

See, that was where I started, but Marcuss has got me to the other end. It is a fair question, though, why does that "some people here have blue eyes" start off the chain reaction, when everyone knows that some people have blue eyes anyway?

It's certainly one I'll want an answer for before I try teaching this one!

I think I see it, as without the trigger there is no expectation for anyone else to commit suicide, but that information creates the expectation that did not otherwise exist.

Here is my attempt to answer my own question. Consider the case of n = 2. When the visitor was not there neither was trying to guess what other is thinking because there was basically nothing to guess about the other Guy's thinking. It's only when the visitor comes they start getting into each other's heads.

I don't think I have articulated that well at all but I think I have it sorted in my mind now

[vcs]

Even in this solution, doesn't it seem kind of strange that they have to wait for 100 days before they can do anything? These are the respective states of mind from Day 1 after the visitor goes:

Blue eyed guys : There are at least 99 other people with blue eyes, and at least 900 other people with brown eyes. So there are two possibilities, I could be the 100th blue-eyed guy, or the 901st brown eyed one.

Brown eyed guys : There are at least 100 other people with blue eyes, and at least 899 other people with brown eyes. So there are two possibilities, I could be the 101st blue-eyed guy, or the 900th brown eyed one.

When they already know so much, why do they have to wait 100 more days to do anything?

[Marcuss]

To find out which one they are! Don't forget they could also be the 1st white/green/yellow/black/lilac

[8ankitj]

Originally Posted by vcs

Even in this solution, doesn't it seem kind of strange that they have to wait for 100 days before they can do anything? These are the respective states of mind from Day 1 after the visitor goes:

Blue eyed guys : There are at least 99 other people with blue eyes, and at least 900 other people with brown eyes. So there are two possibilities, I could be the 100th blue-eyed guy, or the 901st brown eyed one.

Brown eyed guys : There are at least 100 other people with blue eyes, and at least 899 other people with brown eyes. So there are two possibilities, I could be the 101st blue-eyed guy, or the 900th brown eyed one.

When they already know so much, why do they have to wait 100 more days to do anything?

Lol yes. That's another intriguing question