The professions of Mr. Miller, Mr. Painter, Mr. Treasurer and Mr. Hunter are: a miller, a painter, a treasurer and a hunter. However, none of them has a name identifying their profession. They made four statements:
- Mr. Miller is the hunter
- Mr. Treasurer is the miller.
- Mr. Painter is not the hunter.
- Mr. Hunter is not the painter.
According to those statements, the painter must be Mr. Painter, but this cannot be correct. Three of the four statements are false.
Who is the treasurer?
Well, which one of the four statements is true? Let's consider them in turn.
If 1 is true, Mr. Miller is the hunter and the other statements are false.
But if #3 is false, then Mr. Painter is also the hunter.
This is a contradiction, so we know that #1 is not true.
If 2 is true, then Treasurer is the miller, and the other statements are false.
Therefore, from 3, Painter is the hunter, and from 4, Hunter is the painter.
Therefore, Miller is the Treasurer.
This is a legitimate solution. Are there others?
If 3 is instead true, then Painter is not the hunter and the other 3 statements are false.
Miller is not the hunter either, from #1, so Treasurer must be the hunter.
Also, from 4, Hunter is the painter.
Since Miller is not the hunter or the painter, he must be the treasurer.
If 4 is instead true, then Hunter is not the painter, and the other 3 statements are false.
Therefore, from 3, Painter is the hunter.
From 2, Treasurer is not the miller, so he must be the painter.
Since Miller is not the hunter or the painter, he must be the treasurer.
So, we don't know which of the last three statements is true, but in any case Miller is the treasurer.
Final answer.
Analytical Solution (spoiler)
