## I got one.

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

## Do the figures in this diagram look like some sort of hybrid of model rockets and lawn gnomes?

Yes
130
97%
No
4
3%

James
Posts: 74
Joined: Sun Apr 09, 2006 9:28 pm UTC
Contact:

### I got one.

This is one we found long ago, although I just now found this diagram and I hate it.

4men.gif (4.74 KiB) Viewed 9146 times

The four men shown above are buried up to their necks in the ground, and are about to be executed. They cannot move and can only see what is ahead of them. Between A and B is a brick wall which which prevents everyone else from seeing A, and A from seeing anyone else. They know that each of them is wearing a hat, and that two hats are black and two are white, but no one knows which color they are wearing themselves. They will be freed if anyone can determine what color hat they themselves are wearing, all four men will be freed. Also, each person only gets to say either "My hat is white/black," and cannot yell, for example, "Hey, you're wearing a white hat."* However, if this first guess is wrong, they will all be immediately executed. Do the men go free? If so, how?

* Because Kira isn't allowed to cheat.
Last edited by James on Sun Apr 30, 2006 1:30 pm UTC, edited 1 time in total.
This is a block of text that can be added to posts you make. There is a 255 character limit.

kira
I hate bananas.
Posts: 904
Joined: Fri Apr 14, 2006 4:21 am UTC
Location: school
Contact:
i fear i may be confused. couldn't D just yell over to B, "hey, you're wearing a white hat!"?

i guess that'd be cheating. but it doesn't specifically say they can't do that, does it?

James
Posts: 74
Joined: Sun Apr 09, 2006 9:28 pm UTC
Contact:
No, you can't do that. You can only guess as to what color your own hat is. And it says so now, but good point. The guy who typed this up originally left out a lot of detail and added in a lot of stuff you don't need. (eg: model rocket elves.)
This is a block of text that can be added to posts you make. There is a 255 character limit.

Spamtek
Posts: 1
Joined: Sun Apr 30, 2006 2:41 pm UTC

### Sure

Everybody waits a second. Gnome D can't respond, since he can only see one white hat and one black hat, which means there's still a white or black hat he could be wearing. Gnome C realizes this, and since gnome B is wearing a white hat, it means he must be wearing a black one. Gnome C says he's wearing a black hat and they all go free.

Shoofle
Posts: 409
Joined: Sun Apr 09, 2006 9:28 pm UTC
Location: Location, Location.
Contact:
This is the only difficult logic puzzle I know, and I had it explained to me, but I'm still proud that I understand it.

kira
I hate bananas.
Posts: 904
Joined: Fri Apr 14, 2006 4:21 am UTC
Location: school
Contact:
problem. does C know that D is behind him?

because if C can only see what's in front of him and D can't say anything (since he doesn't know what color hat he's wearing), then how does C know that D isn't on the other side of the wall, or that D even exists?

CONUNDRUM.

Shoofle
Posts: 409
Joined: Sun Apr 09, 2006 9:28 pm UTC
Location: Location, Location.
Contact:
Uhhh, C heard them digging the hole for D, or at least D getting into his hole.

xkcd
Site Ninja
Posts: 365
Joined: Sat Apr 08, 2006 8:03 am UTC
Contact:
problem. does C know that D is behind him?

They know the arrangement, yes.

kira
I hate bananas.
Posts: 904
Joined: Fri Apr 14, 2006 4:21 am UTC
Location: school
Contact:
ok, that solves everything. move along, nothing to see here.

Nerd Girl
Posts: 1
Joined: Wed May 31, 2006 5:29 pm UTC
I've been lurking here for some time--the comics are awesome, hey Jello, yadda yadda yadda. But then I stumbled upon something, and I knew I had to come here and show it off:

Sure, the diagram's bad, but it could be worse.

It actually couldn't get worse, as I just stole the image from here because the other one has evaporated - jestingrabbit

Shoofle
Posts: 409
Joined: Sun Apr 09, 2006 9:28 pm UTC
Location: Location, Location.
Contact:
That's... precisely what we have been talking about, with another bad diagram.

xkcd
Site Ninja
Posts: 365
Joined: Sat Apr 08, 2006 8:03 am UTC
Contact:
That's... precisely what we have been talking about, with another bad diagram.

Yeah -- I think that's what she's pointing out.

Shoofle
Posts: 409
Joined: Sun Apr 09, 2006 9:28 pm UTC
Location: Location, Location.
Contact:
Oh! I misunderstood. I actually think that diagram might be a little bit better. It doesn't appear to have the rocket fins.

James
Posts: 74
Joined: Sun Apr 09, 2006 9:28 pm UTC
Contact:
Really though, what are the rocket fins supposed to depict? It that their collars or something?
This is a block of text that can be added to posts you make. There is a 255 character limit.

Shoofle
Posts: 409
Joined: Sun Apr 09, 2006 9:28 pm UTC
Location: Location, Location.
Contact:
My best guess is that they are the sand that is piled up against their necks.

davean
Site Ninja
Posts: 2498
Joined: Sat Apr 08, 2006 7:50 am UTC
Contact:
Since we are debating which is worse, I propose a solution in challenging someone to come up with a universally WORSE diagram, laying to rest the question of which is worse in the sure knowlege that they are both compairably good. I'll be handing out used loli-pops, please collect in person.

James
Posts: 74
Joined: Sun Apr 09, 2006 9:28 pm UTC
Contact:
I think I may have come up with one:

I mean, people's ordering, relative positions, hat color is not only unclear, but oftentimes flat-out misleading. So I guess it could be worse.
This is a block of text that can be added to posts you make. There is a 255 character limit.

xkcd
Site Ninja
Posts: 365
Joined: Sat Apr 08, 2006 8:03 am UTC
Contact:
Impressive!

Shoofle
Posts: 409
Joined: Sun Apr 09, 2006 9:28 pm UTC
Location: Location, Location.
Contact:
Most impressive.

(Haha, I said it!)

davean
Site Ninja
Posts: 2498
Joined: Sat Apr 08, 2006 7:50 am UTC
Contact:
James wrote:I think I may have come up with one:

I mean, people's ordering, relative positions, hat color is not only unclear, but oftentimes flat-out misleading. So I guess it could be worse.

Problem solved! They are both GREAT.

Enzo Dragon
Posts: 4
Joined: Tue Jul 04, 2006 7:16 am UTC
Location: Charon, Pluto's Moon
Contact:

### Re: Sure

Spamtek wrote:Everybody waits a second. Gnome D can't respond, since he can only see one white hat and one black hat, which means there's still a white or black hat he could be wearing. Gnome C realizes this, and since gnome B is wearing a white hat, it means he must be wearing a black one. Gnome C says he's wearing a black hat and they all go free.
Problem! Gnome C can't know that he's wearing a black hat, since he only sees that which is in front of him: the white hat. He could easily be wearing the second white hat, since he can't see D, behind him.

D sees C (black), and B (white), and can only conclude that he could be wearing the second of either color.

C sees B (white), and can only conclude that he could be wearing the second white hat, or either of the two black hats.

B sees no one, and so knows nothing of his hat color.

A sees no one, and so knows nothing of his hat color.

I don't see a solution in any of that information. =/

xkcd
Site Ninja
Posts: 365
Joined: Sat Apr 08, 2006 8:03 am UTC
Contact:
Problem! Gnome C can't know that he's wearing a black hat, since he only sees that which is in front of him: the white hat. He could easily be wearing the second white hat, since he can't see D, behind him.

No; this is the key to the puzzle. If Gnome C were wearing a white hat, Gnome D would know his hat was black and would respond out loud. When Gnome C doesn't hear him respond out loud, he knows he's not wearing a white hat, and his hat is therefore black.

RealGrouchy
Nobody Misses Me As Much As Meaux.
Posts: 6704
Joined: Thu May 18, 2006 7:17 am UTC
Contact:
That assumes that D isn't a wuss.

D could very well see two white hats, but be too scared and/or stupid to conclude that that would mean he has a black hat on.

...also, the third diagram is better because it has colour.

- RG>
Mighty Jalapeno wrote:At least he has the decency to REMOVE THE GAP BETWEEN HIS QUOTES....
Sungura wrote:I don't really miss him. At all. He was pretty grouchy.

Vonkwink
Posts: 12
Joined: Fri Jul 07, 2006 7:57 am UTC
I think to make this question more clear we should add the following like (somewhat borrowed from the blue eyes puzzle): "all gnomes in the puzzle are perfect logicians, and will call out their hat color as soon as they know it."

RealGrouchy
Nobody Misses Me As Much As Meaux.
Posts: 6704
Joined: Thu May 18, 2006 7:17 am UTC
Contact:
These last few posts are just begging for a smartass response, but one isn't coming to me at the moment. Must be the flu.

- RG>
Mighty Jalapeno wrote:At least he has the decency to REMOVE THE GAP BETWEEN HIS QUOTES....
Sungura wrote:I don't really miss him. At all. He was pretty grouchy.

Bahtaw
Posts: 1
Joined: Sun Aug 13, 2006 2:59 am UTC
Location: USA
Achtung!!

Verysillyman
"Do me! Do me!"
Posts: 1442
Joined: Sat Aug 19, 2006 11:25 am UTC
Location: Drinks Cabinet.
Contact:
Baby

brimstone
Posts: 1
Joined: Mon Sep 11, 2006 2:07 pm UTC
Okay, so you have Gnome C saying Black. Because he's relying on a response from the gnome team, and since there's none, he has to answer for D can see B/W can cannot determine his hat, B can't see anything and A can't see anything.

But the solution you guys have only lets gnome C get off free.

Okay, so C is free, you have

A | B D

A is Black, B is White and D is white.

A can't see anything, so obviously he's the last to say his colour from deduction.

There's only 1 Black hat left and 2 whites, but how does D know that he has a white one? He can see that B is white, but cannot see A. So he still has a 50/50 chance of knowing his hat colour right? Did i miss something?

Nevermind, I reread the question and saw that only one guy needed to answer the question right.

I thought that they needed to save themselves...

Marrow
Posts: 101
Joined: Fri Sep 08, 2006 10:24 pm UTC
Location: Au, IL

### Re: Sure

Spamtek wrote:Everybody waits a second. Gnome D can't respond, since he can only see one white hat and one black hat, which means there's still a white or black hat he could be wearing. Gnome C realizes this, and since gnome B is wearing a white hat, it means he must be wearing a black one. Gnome C says he's wearing a black hat and they all go free.

I'll agree that this is the MOST correct solution. It does, however, seem to assume that Gnome C and D know what the other is thinking. I know they can't say what color another gnomes hat is directly, but are they allowed any other form of communication? Because we are looking at them all at once, it would me much harder to find out the answer if the diagram only depicted gnome C's perspective.

If gnome C were able to ask D if he knew what color hat he was wearing and D could say that he did not know without stating the color of B or C's hats then I am perfectly satisfied. Otherwise I am going to whine about the general annoyance I feel for logic puzzles and not bother anyone since I they can just immagine what that is like for themselves.

McMillian
Posts: 27
Joined: Mon Sep 11, 2006 9:09 pm UTC
Contact:

### Re: Sure

Marrow wrote:
Spamtek wrote:Everybody waits a second. Gnome D can't respond, since he can only see one white hat and one black hat, which means there's still a white or black hat he could be wearing. Gnome C realizes this, and since gnome B is wearing a white hat, it means he must be wearing a black one. Gnome C says he's wearing a black hat and they all go free.

I'll agree that this is the MOST correct solution. It does, however, seem to assume that Gnome C and D know what the other is thinking. I know they can't say what color another gnomes hat is directly, but are they allowed any other form of communication? Because we are looking at them all at once, it would me much harder to find out the answer if the diagram only depicted gnome C's perspective.

If gnome C were able to ask D if he knew what color hat he was wearing and D could say that he did not know without stating the color of B or C's hats then I am perfectly satisfied. Otherwise I am going to whine about the general annoyance I feel for logic puzzles and not bother anyone since I they can just immagine what that is like for themselves.
Why don't we just assume that they are all perfect logicians (which should be assumed anyway), and that if there was any way for any of them to know 100% what the color of their hat was, they would instantly say it. As such, when C doesn't hear D instantly say his hat color, he knows that he cannot be wearing the same color hat as the person in front of him because if he was, then D would've known his own hat, and thus C comes to the conclusion his hat is black.

katsielyonz
Posts: 10
Joined: Thu Sep 14, 2006 12:48 am UTC
Location: Bible Belt
I could just be nit-picking but my problem with the solution is that it assumes that Gnome D can see Gnome B. According to the graph all the gnomes are the same exact size, and according to the problem NONE of them can move AT ALL. If that's the case than Gnome D can't see Gnome B and thus Gnome's C's opinion is based on false logic. Gnome D just isn't saying anything because he can't surmise anything from where he's at. It's just that Gnome C has a 50% chance of being right even under false premises.
Yah yah, poo.

McMillian
Posts: 27
Joined: Mon Sep 11, 2006 9:09 pm UTC
Contact:
katsie that's retarded. Would it ease your mind to think that gnome D was bigger than the other gnomes and is further away, so can see B and C but from our perspective looks the same size?

It's a logic riddle, the diagram was just to show you the color pattern

katsielyonz
Posts: 10
Joined: Thu Sep 14, 2006 12:48 am UTC
Location: Bible Belt
Yes, yes it would.

'Sides, there's nothing wrong with creative thinking. I mean without specification how do we know that Gnome C isn't some poor fellow who suffers from Giantism and thus blocks the vision of Gnome D? Either I have to assume the diagram is 100% correct OR I have to allow for the possibility of different sized Gnomes going on OR maybe the executions DIDN'T put them in perfect alignment and so Gnome D CAN see the color of both Gnome's hats; because the word problem says specifically they can't move at all.
Yah yah, poo.

Marlayna
Posts: 104
Joined: Sun Sep 10, 2006 10:43 am UTC
So, they're all perfect logicians, but does each one know that the others are perfect logicians?

Boy oh boy I love nitpicking
There are 10 kinds of people.
Those who can read binary numbers and those who can't.

katsielyonz
Posts: 10
Joined: Thu Sep 14, 2006 12:48 am UTC
Location: Bible Belt
Whoa, I didn't even think about that nitpick.
Yah yah, poo.

Marrow
Posts: 101
Joined: Fri Sep 08, 2006 10:24 pm UTC
Location: Au, IL
Actually only gnome C has to be the perfect logician, as long as A, B and D are not stupid C has a good chance of figuring things out before the others do something to condemn them all. However this is not something one should expect from gnomes since they are being put to death in the first place they can't be all too genius.

Verysillyman
"Do me! Do me!"
Posts: 1442
Joined: Sat Aug 19, 2006 11:25 am UTC
Location: Drinks Cabinet.
Contact:
This is in a country where is it illegal to be a perfect logician, especialy if you look like a rocketship. That's why they're being put to death.

Teaspoon
Posts: 351
Joined: Tue Sep 12, 2006 11:37 pm UTC
Location: Where you least expect me
Of course. Nobody likes a smartarse.

Binary.Tobis
Posts: 34
Joined: Mon Sep 25, 2006 4:35 pm UTC
I feel silly for not getting the answer sooner.
"A diplomat is a man who always remembers a woman's birthday, but never remembers her age."

ch103
Posts: 1
Joined: Tue Oct 23, 2007 6:14 am UTC

### Re: I got one.

C calls out that he is wearing a black hat. Why is he 100% certain of the colour of his hat ?

After a while, C comes to the realization that he must answer.

This is because D can't answer, and neither can A or B.

D can see C and B, but can't determine his own hat colour. B can't see anyone and also can't determine his own hat colour. A is in the same situation as B, where he can't see anyone and can't determine his own hat colour.

Since A, B, and D are silent, that leaves C. C knows he is wearing a black hat because if D saw that both B and C were wearing white hats, then he would have answered. But since D is silent, C knows that he must be wearing a black hat as he can see that B is wearing a white hat.