The game completes inPreamble: 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
In order for 7 hats to be on the following caseslogicians 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:
12222
$L_1$ sees 8 hats and can immediately deduce "I have one hat on my head."
Each following logician surmises that theyFrom the viewpoint of the other logicians, $L_1$ must have 2 hats because theseen zero 1-hatted logicians. Had he seen one, $L_1$ was confidentwould not have enough information to state his hat result immediatelydeduce 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.
The game does not complete in the following casesCase 21222:
$L_1$ is wearing 2 hats. The logician immediately before the one with 1 hat is unable to guess his hat count. Note: Proof to follow later. This work left in temporarily.
21222
$L_1$ sees 7 hats. He cannot determine if he has 1 or 2 hats.
$L_1$ could never see 4 hats. The maximum at the table would thencould be 6; not enough hats.
arranged as 11222 or 21222 from the perspective of $L_1$ could not have seen 5 hats. Had he, he would know he had 2 hats.
$L_1$ couldIt is unknown to the logicians at this point that 7 is not have seen 8 hatsa valid hat count. Had heAs such, he$L_1$ would knowalso not be able to determine if he hadsaw two 1 hat-hatted logicians.
$\therefore$ All of the other logicians know thatSo at this point, $L_1$ must have seen 6 or 7 hats$L_{3,4,5}$ 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 $L_2$ could not have seen 6 hats. Had he,saw 1 hat on any of $L_3$$L_{3,4,5}$, he would determinebe unable to deduce that he isn'tis wearing a1 hat. It is at allthis point that (which is impossible) since$L_{3,4,5}$ know they are each wearing 2 hats.
If $L_2$ saw 1 hat on $L_1$, he would still be able to make his statement, so $L_1$ is not able to guess and the game does not complete.
Case 22122:
$L_1$ sees 67 hats on three of. He cannot determine if he has 1 or 2 hats. The table could be arranged as 12122 or 22122 from the same peopleperspective of $L_1$.
The only way thatFrom the perspective of $L_2$$L_{2,4,5}$, the table could have seen 7 hats andadditionally be set up as 2x122, 221x2, 2212x respectively.
$L_2$ having no new information cannot make a confident statement about his own numberdecision when seeing only one hat.
$L_3$ of hats is to seecourse knows that he's wearing 1 hat because he sees 8.
Again, if $L_3$ saw 1 hat on any of $L_1$$L_{1,4,5}$, he would be unable to deduce that he is wearing 1 hat. InIt is at this case,point that $L_{3,4,5}$$L_{1,4,5}$ know they are each wearing 2 hats.
$\therefore$If $L_{3,4,5}$ know that$L_3$ saw 1 hat on $L_2$ saw, 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), butso $L_1$$L_2$ is unsurenot able to guess and the game does not complete.
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22122
Case 22212:
$L_1$ same as the 21222 scenario.
$L_2$ sees 7 hats. He counts 5 the samecannot determine if he has 1 or 2 hats. The table could be arranged as 12122 or 22122 from the perspective of $L_1$ and two on.
From the perspective of $L_1$ 's head$L_{2,3,5}$, the table could additionally be set up as 2x212, 22x12, 2221x respectively.
$L_2$ could have seen 6 or 7 hatshaving 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.
Suppose $L_2$ sees 6 hats. This implies that $L_3$ countsAgain, if $L_{1,4,5}$ as two 2-hatted logicians and one$L_4$ saw 1-hatted logician. Each hat on any of $L_{1,4,5}$ see two 2-hatted logicians$L_{1,2,5}$, so might concludehe would be unable to deduce that he is wearing 1 hat. It is at this point that $L_{1,2,5}$ know they are the oneeach wearing 2 hats.
If $L_4$ saw 1 hat. However, on $L_3$, he would havestill be able to consider the possibility thatmake his statement because he knows $L_2$ saw 7$L_3$ was unable to guess (i.e., he knows $L_3$ did not see 8 hats), so would$L_3$ is not be able to definitively say thatguess and the game does not complete.
Case 22221:
$L_1$ sees 7 hats. He cannot determine if he was wearinghas 1 hator 2 hats. The table could be arranged as 12122 or 22122 from the perspective of $L_1$.
$\therefore$ In order forFrom 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$ to statehaving 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 ishe's wearing 1 hat, because he must have counted 6 hatssees 8.
Again, if $L_5$ saw 1 hat on any of $L_{1,4,5}$$L_{1,2,3}$, placing two hats on each headhe would be unable to deduce that he is wearing 1 hat.
$L_{1,4,5}$ can now all say 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), butso $L_2$$L_4$ is still unsurenot able to guess and the game does not complete.
It's a lot of work to continue, so I'll leave the rest for later.
