Can You Solve The 4 Hats Logic Puzzle?

Four friends claim they have been wrongly imprisoned. They are not criminals but mere harmless logicians! The warden says, "If you are in fact logicians I will set you free. But first you have to prove it to me." Can they work it out?

I have seen this puzzle shared in many places. I based my graphic on a widely circulated graphic but I have been unable to determine the original source–if you know it please let me know. The links below are examples of how the puzzle has been shared.

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[londonbobby]

If Logician 1 sees two hats the same colour then he knows his hat is the opposite colour and speaks out. After a while of not speaking out Logician 2 realises that 1 must be looking at two different colours and therefore his hat is the opposite colour to the one in front of him and he speaks out. Basically it's the same, but without the need for the logic table.

[ongbonga]

Number 1 isn't confused, he's a perfect logician, he's impatiently waiting for 2 to hurry up and get the message.

[matthewmitchell]

Three logicians walk into a bar. The bartender asks them "Would you three like something to drink?" The first logician says "All three of us? I don't know." The second logician says "I don't know either." The third says "Yes, we would."

[IoEstasCedonta]

...I feel like you've made this more complicated than it had to be.

[dale]

You don't really need to enumerate all 6 possibilities in this case. If the 2nd and 3rd person wears the same color, then the 1st person can immediately answer. If they wear different color, then the 1st will stay silent. And that silence becomes information to the 2nd person, who knows the color of the 3rd and, hence, also knows his. I think this concept is called common knowledge

[mytube]

I misheard the rules. I thought all four had to state the color of their own hat for them to be set free, and I just couldn't see that happening.

[TheOneAndOnlyCatfish.]

these guys cannot communicate with eachother.
Proceeds to make them communicate.

[katiekawaii]

It could be expressed more simply. If #1 sees two of the same color hat, he says he has the other one. If #1 says nothing, then #2 knows he and the guy in front of him have different color hats, so he says the opposite of whatever color he sees on the guy in front of him.

[OnlyPenguian]

Let's assume that "they are not allowed to talk to each other" means that they can't communicate to each other at all. Then the only way that #2 would know that #1 was uncertain was via the fact that the warden has not already set everyone free. The problem is therefore better posed if there are time constraints.

[kayr]

You made this so much more convoluted than it needed to be

[siarheipisarau]

From the problem statement, it is not clear that the first person can see the next two hats. It is really important to know this information for problem solving

[Wou_]

That was... an unnecessarily long explanation. As other people pointed out, if they know there are 2 red hats and 2 blue hats (which is the case according to the sources in description), logician 1 will have to remain silent, as seeing different hat colors on 2 and 3 means he could have either color. Then logician 2 notices 1's silence, and thinks, "If 3 and I had the same color, 1 would immediately know his own hat color. As this was not the case, 3 and I have different hat colors, meaning that since 3 has a red hat, my own hat is blue".

[mr.l]

I think firmly stating that they can't communicate in any way was a little confusing here. If we had clarified that there are clock ticks, and on each tick a logician can answer (and all other logicians take note of when no-one else answers), then it would've made it more clear that the passage of time could be used to communicate whether a logician does/doesn't know the answer. It's a pattern used in other hat style logic puzzles so it's not too far out of reach to imagine, but it's worth clarifying.

e.g.: the blue-eyed people on an island logic puzzle uses the concept of "days" to let the puzzler understand that the passage of time is divided into distinct rounds, which can tell agents at which point other agents were uncertain of the answer.

[matthewharrigan]

The obvious answer is to look at the brim of your own hat

[markburgin]

What a convoluted way to explain a simple solution.

[Maxm]

Surprisingly, 3 is the next person that can guess his hat, after 2 shouted blue then if 3 is blue, 1 would shout right away, so 3 knows his hat is red

[b_z]

You should include a timeframe for when they answer to make the puzzle more clear something like every 5 minutes the warden will ask for an answer so that the time between isn’t arbitrary

[SirKenchalot]

I didn't iterate the possible arrangements but just worked out that in general, Logician 4 can see 2 logicians and if they have different colored hats, he cannot be certain of the color of his own hat. So, given Logician 1's uncertainty, Logician 2 knows his hat must be the opposite color to that which he can see on Logician 3, hence he announces he is wearing a blue hat.

[ludo]

You give the rules that they cannot communicate to each other, but then says #2 knows their hat because they have the information that #1 see 2 different colors. But without communicating he cannot know that …

[blueyindustries]

I misheard the instructions at first and try figuring out how they could all state the color of the hat for certain. I’m glad I listened to the instructions again.