Brain Teasers and Maths Puzzles

[Pater Alf]

I have to admit that I had to make a small drawing to not get confused by my own logic .

[Zyx]

My second favorite brain teaser is the one that is asked before the final boss fight in Kingdom of Loathing. The developers being quite huge dicks, the puzzle is of course a variation of the aptly named The Hardest Logic Puzzle Ever. For those who don't know this browser based game, the point in the game is to play it through multiple times so you would at each playthrough asked to complete this puzzle. Naturally it was randomized each time to be a bit different.

[eMTe]

Both are based on mathematical logic though. This one is based on pure logic:

A businessman hired an old pal from school as a night guard in his small firm. Everything was ok for several months, guard was fulfilling his duties, but suddenly, one day before his employer's important business trip he came to him with a disturbing message.
-Listen - he said - last night I had a dream that the plane you're going to fly will crash and there will be no survivors. Postpone your trip if you can.
The businessman thought "well, he's always been a weirdo, but what if this dream was prophetic?" and rescheduled the trip. Next day he was watching the evening news and - surprise - the plane crashed indeed, leaving no survivors. Next day he went to his clairvoyant friend to thank him, but after that he said: "Unfortunately, I have to fire you".

Why?

[Pater Alf]

That one's easy: Because he slept on duty.

[Drasir-Vel]

My second favorite brain teaser is the one that is asked before the final boss fight in Kingdom of Loathing. The developers being quite huge dicks, the puzzle is of course a variation of the aptly named The Hardest Logic Puzzle Ever. For those who don't know this browser based game, the point in the game is to play it through multiple times so you would at each playthrough asked to complete this puzzle. Naturally it was randomized each time to be a bit different.

Reminds me of this one with Hitler Stalin and Goebbels.
http://goodtoknowit.blogspot.com/2007/0 ... 4av2FHxWDE

[Zyx]

There is a scientifically studied way to make these problems easier to solve.

[Chroelle]

Yeah that one has been getting alot of praise by my male friends today. It was in a Danish paper, and was the talk of the town at my workplace.

[eMTe]

"people who drank alcohol and had a blood alcohol level of 0.07 or higher were worse at completing problems that required attentional control but better at creative problem solving tests"

"people who were allowed to sleep after being given a problem were also more likely to come up with a creative solution compared to those who stayed awake"

I wonder if we really need a bunch of US scientists to spread revelations that are obvious to common man for ages.

[Drasir-Vel]

Reminds me of this.

[Tormuse]

...And the puzzles remind me of this comic.

EDIT: Also, the fact that the above article says "most" doctors makes me wonder about the doctors who disagree!

[Zyx]

...And the puzzles remind me of this comic.

...which in turn reminds me of this logic puzzle. (getting back to topic)

[eMTe]

Another hardest puzzle. :/

I can see that it is probably fine from the logic side, but it is written in a very messy manner, so I dont understand it. Ferry visits the island at noon every day, right? At this very moment Guru speaks that she sees somebody with blue eyes (I presume she tells the other islanders about someone who guessed the colour of his own eyes and can leave the island, right?). What happens if all people will leave the island? Does Guru take part in the game and can deduce the colour of her eyes? It is indeed one of the hardest puzzle, if we agree that written language also should follow some logic.

[Tormuse]

...For those who don't know this browser based game, the point in the game is to play it through multiple times so you would at each playthrough asked to complete this puzzle...

I still want to give this one a try. (I admit that I accidentally saw part of the answer because I didn't realize it was written on the same page, but since no one else has posted it, I'll share my own deduction)

Okay, so the first thing I notice is that west keeps saying "Graaaaagh," which suggests that he's the one who craves human flesh and can safely be ignored. (Or at least his responses can be ignored)

West craves human flesh

Of the other three, one tells the truth, one lies, and one alternates between telling truth and lies.

The next thing I notice is that North contradicts himself, which means he is either the one who always lies or sometimes lies. Let's assume, for a moment, that he's the one who sometimes lies, then that means that South and East are the ones who always tell the truth and always lie. (which one is which, we don't know) Now, in the first three statements, North says that South is lying and East says that North is lying. If South was, in fact telling the truth, then that would mean North was lying, which would mean that East was also telling the truth. South and East can't both be telling the truth, so that doesn't work. If South was lying, then that means North was telling the truth, which would mean East was lying. South and East can't both be lying, so that doesn't work either. Since none of those combinations work, that means my initial assumption that North is the one who sometimes lies was wrong, and so he must be the one who always lies!

tl;dr: North always lies

So, between South and East, one of them always tells the truth and the other alternates between lying and telling the truth. Since North said that South is lying and we know that North always lies, South must have been telling the truth! And since East said that North was lying, East was also telling the truth. So, whichever one of them is the alternator told the truth this time, so his next statement will be a lie. That means that when they tell the first and third digits, one is lying and the other is telling the truth, and when they say the second digit, they're both telling the truth...

After that, I'm stumped... All I know for sure is that the second digit is 4. I have to go to work now, but I'll give this some more thought when I get back home. (I'm trying not to look at any more of the answer)

EDIT: Okay, I'm back to finish the puzzle.

Both North and South say that the third digit is 5. Since we know North is lying, that means South is lying too. That means that East is the one telling the truth.

Therefore, the combination is the one that East gives, 842.

EDIT2: I just looked at the answer posted in the link... I don't know who posted that, but there are flaws in their logic and they ultimately came to the wrong answer. I like my answer better.





As for the comic I posted, you can always find out who is who by asking "Are you the one who stabs people?" The one who tells the truth will say "no." The one who lies will say "yes." The one who stabs people will just stab you. The drawback is that one out of three times you do this, you will be stabbed.

[Zyx]

I can see that it is probably fine from the logic side, but it is written in a very messy manner, so I dont understand it. Ferry visits the island at noon every day, right?

The ferry visits the island at midnight. At that point, any islander who has figured out his/her eye color, leaves the island.

At this very moment Guru speaks that she sees somebody with blue eyes (I presume she tells the other islanders about someone who guessed the colour of his own eyes and can leave the island, right?).

No. On one day, (at around noon-ish), she says once "I can see someone who has blue eyes.". After that she is silent.

What happens if all people will leave the island?

Then all people leave the island.

Does Guru take part in the game and can deduce the colour of her eyes?

She is an islander so she is part of the game and can possibly deduce the colour of her eyes. If she does, then she leaves the island.

It is indeed one of the hardest puzzle, if we agree that written language also should follow some logic.

I think it's pretty clearly stated.

[Zyx]

As for the comic I posted, you can always find out who is who by asking "Are you the one who stabs people?" The one who tells the truth will say "no." The one who lies will say "yes." The one who stabs people will just stab you. The drawback is that one out of three times you do this, you will be stabbed.

You apparently didn't notice the hover text "And the whole setup is just a trap to capture escaping logicians. None of the doors actually lead out." =)

[Tormuse]

I did notice. You apparently didn't notice that I didn't say it would help you get out.

I just edited my other post to complete my answer to the doors puzzle. Now, I'll take a look at the other ones.

EDIT: As for the Blue Eyes puzzle... my first thought is that there's some trick to it, like the Guru has her back to most of the islanders or most of them have their eyes closed or something like that, but the instructions for the puzzle say that there are no tricks and that it isn't a lateral thinking puzzle, so that seems to rule out those possibilities. Beyond that, I don't know... it explicitly states that "Everyone can see everyone else at all times," so that seems to make the statement of "I can see someone who has blue eyes," meaningless. I'll give it some more thought, but if the answer isn't "no one leaves," then I don't see how you could deduce any answer from that.

EDIT2: I don't know if this is the right answer, but... is there any penalty to the islanders for a wrong answer? If not, and the ferry comes every night, then they might as well just guess every night. What's to stop all 201 of them from going to the ferry and saying, "I have blue eyes?" Then, all the blue-eyed ones would leave. The night after that, with all the blue-eyed ones gone, they can safely guess that they all have brown eyes and all the brown-eyed ones would leave. Unfortunately for the Guru, she'll be left behind, because there's no one to tell her that she has green eyes and no way for her to know that she's different from everyone else. Of course, if this is the correct answer, then that means the Guru's statement in the puzzle really is meaningless! They could all have deduced their eye colours years ago by this method without the need of the Guru!

EDIT3: Never mind, it says they didn't guess. I think I need a hint now.

[Zyx]

it explicitly states that "Everyone can see everyone else at all times," so that seems to make the statement of "I can see someone who has blue eyes," meaningless. I'll give it some more thought, but if the answer isn't "no one leaves," then I don't see how you could deduce any answer from that.

Notice that no-one knows their eye color. They do not know that the options are either blue, brown or green. Before the Guru spoke, any person could have thought their eyes are red, but could see 1 green eyes, and 99 blue and 100 brown or 100 blue and 99 brown eyes.

What's to stop all 201 of them from going to the ferry and saying, "I have blue eyes?" Then, all the blue-eyed ones would leave.

I'm going to highlight this here for no reason whatsoever.

The night after that, with all the blue-eyed ones gone, they can safely guess that they all have brown eyes and all the brown-eyed ones would leave.

This would not happen because they do not know that all the only options are brown, green and blue. They can't deduce from "all blue eyes are gone and I can only see brown eyes, and wait a sec, and one set of green eyes, so ... oh wait"

They could all have deduced their eye colours years ago by this method without the need of the Guru!
EDIT3: Never mind, it says they didn't guess. I think I need a hint now.

Hint: Consider the case of 100 brown eyes, 1 blue eyes and the Guru. The Guru says "I can see a person with blue eyes". What happens?

[Tormuse]

Hint: Consider the case of 100 brown eyes, 1 blue eyes and the Guru. The Guru says "I can see a person with blue eyes". What happens?

Okay... In that case, the one guy with blue eyes would look around, see that no one else has blue eyes and deduce that he's the one with blue eyes and he would leave... Similarly, if the Guru had said, "I see 100 people with blue eyes," the blue-eyed people would look around and see 99 people with blue eyes and each and every one of the blue-eyed people would deduce that they have blue eyes and leave...

but since she *didn't* say that, her statement is meaningless, because each of the blue-eyed people will look around and assume that she's talking about the other blue-eyed people. The puzzle states that this is the only thing she has *ever* said, so there's nothing that would have narrowed their numbers down to one.

Each blue-eyed person will see 99 blue-eyed people. Each brown-eyed person will see 100 blue-eyed people. Since there's (apparently) no reason for them to assume anything about the actual numbers, and the Guru said nothing about numbers, there's simply not enough information for them to deduce anything! No individual has any way of knowing how he relates to the total number of blue-eyed people because there has been no reference to the number in what the Guru said.

Either you're a blue-eyed person and you see 99 blue-eyeds, 100 brown-eyeds, and 1 green-eyed... or you're a brown-eyed person and you see 100 blue-eyeds, 99 brown-eyeds, and 1 green-eyed... the only exception is the Guru herself who sees 100 blue-eyeds, 100 brown-eyeds, and 0 green-eyeds.

"I can see someone who has blue eyes."

I don't see how any of them can deduce any numbers, and therefore their own eye colour from this. This puzzle had better have one hell of a clever answer!

[Zyx]

Okay... In that case, the one guy with blue eyes would look around, see that no one else has blue eyes and deduce that he's the one with blue eyes and he would leave...

Correct.

I don't see how any of them can deduce any numbers, and therefore their own eye colour from this.

Okay, second hint: Consider the case of 2 blue-eyed persons and 100 brown-eyed persons and the Guru. What happens?

This puzzle had better have one hell of a clever answer!

Oh, it does.

[Tormuse]

I don't see how any of them can deduce any numbers, and therefore their own eye colour from this.

Okay, second hint: Consider the case of 2 blue-eyed persons and 100 brown-eyed persons and the Guru. What happens?

Okay, I think I see where you are going... Each of the two blue-eyeds would look at each other and assume that the Guru is referring to the other guy. Then, after the first night passes and neither of them has left, they will both realize that the other guy didn't leave because they made the same assumption. Then, both of them will leave on the second night.

That works fine for two blue-eyeds, but I don't see how that can be extended to 100 blue-eyeds. Each blue-eyed person will assume that the Guru is referring to one of the other 99 people. There's no reason for any individual blue-eyed person to come to the conclusion that they are one of the 100 blue-eyed people, any more than there is any reason for one of the brown-eyed people to come to the conclusion that they are one of 101 blue-eyed people. (Since they don't know their total numbers) It doesn't matter how many nights pass; each night, no one will leave, each blue-eyed person will always see 99 other blue-eyed people, and they will always continually assume that no one left because, with so many blue-eyed people around, they all know that they can't guess their own eye colour from the Guru's vague statement.

Sorry, Zyx. I still don't see how I can reach an answer from this.