4 ms·
I think that, yes, on her turn, she did know that I knew that she knew that I could not see 20. Postulate 1: Total number of statues are 18 or 20 Lemma
by topmonk 8y ago
I think that, yes, on her turn, she did know that I knew that she knew that I could not see 20.
Postulate 1: Total number of statues are 18 or 20
Lemma 1, she knows that I can either see 10 or 12 statues.
Proof: She saw 8 statues. Since the total is either 20 or 18, it
means that she knows I can either see 10 or 12 by Postulate 1
Lemma 2, suppose she saw 10 statues, she knows that I did not see 20.
Proof: Assume she saw 10 statues. Since the total is either 20 or
18, it means that she knows I can either see 8 or 10 by Postulate 1
Lemma 3, suppose she saw 6 statues, she knows that I did not see 20.
Proof: Assume she saw 6 statues. Since the total is either 20 or 18,
it means that she knows I can either see 14 or 12 by Postulate 1
Prop: She knows that I knew that she knew that I could not see 20.
Proof:
She knows I can see 10 or 12. (Lemma 1)
Lets consider the first case, from her viewpoint, that I have 10 statues.
If I saw 10 statues, then I know she can only have seen 8 or 10
statues, by Postulate 1
If she saw 8, then by Lemma 1, she knew I could not see 20.
If she saw 10 statues, then by Lemma 2, she also knew I could
not see 20.
Therefore, if I have 10 statues, I know that she knows that I
could not see 20.
Now for the second case, from her viewpoint, that I have 12 statues.
If I saw 12 statues, then I know she can only have seen 6 or 8
statues, by Postulate 1
If she saw 8, then by Lemma 1, she knew I could not see 20.
If she saw 6 statues, then by Lemma 3, she knows I could not see
20.
Therefore, if I have 12 statues, I know that she knows that I
could not see 20.
For all the cases she knows that I could have, I know that she knows
that I could not see 20.
Therefore she knows that I know that she knows that I could not see
20.
QED.
- thechao 8y agoI don't care to figure out if this complicates any solution schema, but your Lemma 1 is wrong. If I can see 8 statues, then my friend can see anywhere from 12 to 20 statues --- at no point does the riddle say that we see 18 or 20 unique statues. It's the imprecision in these sorts of riddles when told, but which expect extraordinary precision in the solver, that generally turns me off from the problem.
- another_one_off 8y ago"Out of your friend’s window, which overlooks the opposite side of the graveyard, she can see eight." The unique is (poorly) implied, imo.
- thechao 8y agoJust to play the devil’s advocate: I imagined looking at the left side, and my friend was looking at the opposite side, ie, the right side.
- topmonk 8y agoIf my Lemma 1 is wrong, then the solution proposed by the article is also incorrect, because it also assumes this.
- mfoy_ 8y agoThe problem is confusing. You are looking out on different sides of the graveyard, so you know there is no overlap in statues, and the caretaker informs you that the total number of statues is either 18 or 20. So you know your friend can only see either 6 or 8. I agree it's not super well-phrased though.
- grraaaaahhh 8y agoYes, but there is some level of meta-knowledge that you're eliminating. She sees there are 8 statues. She knows that you see either 10 or 12. She knows that you know that she sees either 8 or 6. She knows that you know that she knows that you see either 10, 12 or 14. She knows that you know that she knows that you know that she sees either 8, 6 or 4. ...etc. That's why the solution takes 5 days, each day eliminates another level of meta-knowledge until you get to a point where you know how many statues the other must be seeing based on how many times they passed.