Replying to Avatar Hanshan

But what information does he use to deduce that 1 and 2 must have a beer?

If all 3 enter into the joke in the "not-knowing" state [ 0 0 0 ]

then through the dialogue, information must be imparted to make it [ 0 0 1 ]

but they say "I don't know"

which afaict doesn't give 3 any information to make a deduction.

(not to over-intellectualize your joke or anything)
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Hanshan 3mo ago

sorry

that set is [ 0 0 1 ] already

since 3 knows whether he want a beer or not,

but he needs to determine which of the other four possible configurations exist.

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YODL 3mo ago

If [0 y z] then 1 would be able to say no on his turn, since him being 0 is sufficient to reply to question that way (he knows at least one person, himself, will not have beer). If [1 y z] he would not know for sure. Thus we know it's 1 for him on turn 2.

Similarly for 2.

So three knows it's [1 1 z], and we learn he also wants a beer when he answers yes.

The side joke I shared is actually more clear on some respects, if that helps (even though was meant as insult to npub Techno).

Hope that helps