Who broke the window?
Mary has 4 children: Alice, Brian, Casey and David. Once Mary left the kids alone in the house. When she returned, she found a window broken, and there were 4 notes left on the table:
- I've broken the window. The author of the 4 note is lying.
- Brian is lying today, but he hasn't broken the window.
- I'm Brian and I'm telling the truth today. The author of the 2 note has broken the window.
- David has written the 1 note, and Casey has broken the window.
Mary knows that Alice and Casey never lie, Brian tells the truth with 50% probability (see below for more about Brian), and David always lies.
Who broke the window?
Details and Assumptions:
- All Brian's words in his note are either a lie or a truth, with equal chances.
- Each kid has written exactly 1 note.
- Mary can't identify her kids by their handwriting.
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Note 1 & 4 are flinging accusations towards each other, so one of them is a truthful note and the other a lying note. One the other hand, Note 2 also accuses a named Brian, who the author of Note 3 admitted to be, and therefore its author would either involved in a confrontation with Note 2 if the admission is true (and again, the same truthful note vs lying note like we've seen between Note 1 & 4 before), or a liar by their own right if it's not. Since we can only find 2 truthful notes at most, it must be that both of the boys lied and there's a girl writing either Note 2 or 3. Obviously, this honest girl wouldn't self-identify herself as Brian, so she must be the writer of a truth telling Note 2.
As stated by Note 2 (and the notes from Details & Assumptions), Note 3 must have been written by David, and furthermore, Note 4 by Brian. This leave Note 1 to be authored by one of the girls, who can't be Casey as accused by her lying brother, Brian. As such, Alice wrote Note 1 and Casey wrote Note 2.
By her own admission, Alice plead guilty to the case of the broken window.