What I'm left wondering is, if they all trust each other, and they can see each others hats, could they not just tell each other? As nice as logical observation is, in this example, because there are three of them, any one could have the other two agree on a colour, so the peer review process seems a preferable solution.
Because it is unimaginably rude to speak in the presence of the king without him adressing you first. The rules of etiquette forbid it.
The Millennium school has 1000 students and 1000 student lockers. The lockers are in a line in a long corridor and are numbered from 1 to 1000.
Initially all the lockers are closed (but unlocked).
The first student walks along the corridor and opens every locker.
The second student then walks along the corridor and closes every second locker, i.e. closes lockers 2, 4, 6, etc. At that point there are 500 lockers that are open and 500 that are closed.
The third student then walks along the corridor, changing the state of every third locker.
Thus s/he closes locker 3 (which had been left open by the first student), opens locker 6 (closed by the second student), closes locker 9, etc.
All the remaining students now walk by in order, with the nth student changing the state of every nth locker, and this continues until all 1000 students have walked along the corridor.
(i) How many lockers are closed immediately after the third student has walked along the corridor?
(ii) How many lockers are closed immediately after the fourth student has walked along the corridor?
(iii) After the hundredth student has walked along the corridor, what is the state of locker 1000?
(iv) At the end (after all 1000 students have passed), which lockers are open and which are closed?