There are five logicians seated around a table. They are blindfolded and hats are placed on their heads. After removing the blindfolds, they are told a true statement:
Each of you has either one hat or two hats on his head. The total number of hats is 7, 8 or 9.
The actual number is 9, but this is not told to them.
They are then told to answer the question "How many hats are on your head?" in clockwise fashion (starting from any of the 5). If it is possible to logically deduce the number of hats on their own heads, they do so, otherwise reply "I don't know." and wait till the question cycles back to them.
They have no means of knowing the number of hats on their own heads (except logically), but can see every other person. They have no other means of communication, and no tricks involved.
At what number should the one-hatted logician be placed so that:
- The game is completed?
- The game goes on forever?
Note
I made a small error while posting the question. Since people have anyway up-voted and answered this one, I've left it as it is and posted the actual question as a second version (click here to view)
Answer
Preamble: The following are all of the possible (unordered) hat combinations that could exist for a total of 7, 8, or 9 hats:
22221: 9 hats
22211: 8 hats
22111: 7 hats
TL;DR
The game only completes when the logician with 1 hat is asked first.
In order for 7 hats to be on the logicians heads, none of the logicians can see three 2-hatted logicians. All of the logicians see at least three 2-hatted logicians, so 7 hats is not possible. This leaves the logicians considering only the 8- and 9-hat scenarios.
Case 12222:
$L_1$ sees 8 hats and can immediately deduce "I have one hat on my head."
From the viewpoint of the other logicians, $L_1$ must have seen zero 1-hatted logicians. Had he seen one, $L_1$ would not have enough information to deduce whether this was an 8-hat or 9-hat scenario.
$\therefore L_{2,3,4,5}$ are all able to deduce that they each wear 2 hats and the game completes.
Case 21222:
$L_1$ sees 7 hats. He cannot determine if he has 1 or 2 hats. The table could be arranged as 11222 or 21222 from the perspective of $L_1$.
It is unknown to the logicians at this point that 7 is not a valid hat count. As such, $L_1$ would also not be able to determine if he saw two 1-hatted logicians.
So at this point, $L_{3,4,5}$ see the table as 21x22, 212x2, 2122x respectively, where x is still unknown.
$L_2$ sees 8 hats and can determine that he has 1 hat.
If $L_2$ saw 1 hat on any of $L_{3,4,5}$, he would be unable to deduce that he is wearing 1 hat. It is at this point that $L_{3,4,5}$ know they are each wearing 2 hats.
If $L_2$ saw 1 hat on $L_1$, he would still be able to make his statement, so $L_1$ is not able to guess and the game does not complete.
Case 22122:
$L_1$ sees 7 hats. He cannot determine if he has 1 or 2 hats. The table could be arranged as 12122 or 22122 from the perspective of $L_1$.
From the perspective of $L_{2,4,5}$, the table could additionally be set up as 2x122, 221x2, 2212x respectively.
$L_2$ having no new information cannot make a decision when seeing only one hat.
$L_3$ of course knows that he's wearing 1 hat because he sees 8.
Again, if $L_3$ saw 1 hat on either of $L_{4,5}$, he would be unable to deduce that he is wearing 1 hat. It is at this point that $L_{4,5}$ know they are each wearing 2 hats.
If $L_3$ saw 1 hat on $L_2$, he would still be able to make his statement because he knows $L_2$ was unable to guess (i.e., he knows $L_2$ did not see 8 hats), so $L_2$ is not able to guess and the game does not complete.
Case 22212:
$L_1$ sees 7 hats. He cannot determine if he has 1 or 2 hats. The table could be arranged as 12122 or 22122 from the perspective of $L_1$.
From the perspective of $L_{2,3,5}$, the table could additionally be set up as 2x212, 22x12, 2221x respectively.
$L_2$ having no new information cannot make a decision when seeing only one hat.
$L_3$ having no new information cannot make a decision when seeing only one hat.
$L_4$ of course knows that he's wearing 1 hat because he sees 8.
Again, if $L_4$ saw 1 hat on any of $L_{1,2,5}$, he would be unable to deduce that he is wearing 1 hat. It is at this point that $L_{1,2,5}$ know they are each wearing 2 hats.
If $L_4$ saw 1 hat on $L_3$, he would still be able to make his statement because he knows $L_3$ was unable to guess (i.e., he knows $L_3$ did not see 8 hats), so $L_3$ is not able to guess and the game does not complete.
Case 22221:
$L_1$ sees 7 hats. He cannot determine if he has 1 or 2 hats. The table could be arranged as 12122 or 22122 from the perspective of $L_1$.
From the perspective of $L_{2,3,5}$, the table could additionally be set up as 2x212, 22x12, 2221x respectively.
$L_2$ having no new information cannot make a decision when seeing only one hat.
$L_3$ having no new information cannot make a decision when seeing only one hat.
$L_4$ having no new information cannot make a decision when seeing only one hat.
$L_5$ of course knows that he's wearing 1 hat because he sees 8.
Again, if $L_5$ saw 1 hat on any of $L_{1,2,3}$, he would be unable to deduce that he is wearing 1 hat. It is at this point that $L_{1,2,3}$ know they are each wearing 2 hats.
If $L_5$ saw 1 hat on $L_4$, he would still be able to make his statement because he knows $L_4$ was unable to guess (i.e., he knows $L_4$ did not see 8 hats), so $L_4$ is not able to guess and the game does not complete.
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