Burglars and Detectives Part1

If Alice is telling the truth, that means that Carl is lying and he took the money. However, if Carl is lying, then that means that Alice did take the money. The problem says that only one of the three people took the money, so this is impossible. Alice must be lying.

If Alice is lying, that means that Carl is telling the truth and did not take the money. If Carl is telling the truth, then Beto is lying and Alice didn't take the money. If Beto is lying, then Carl did not take the money, but Beto did. There are no contradictions here, so Beto took the money.

Notice that either Alice or Carl is lying, because Alice said Carl is lying.

Also, notice that either Carl or Beto is lying, because Carl said Beto is lying.

Finally, notice that both Alice and Beto are either telling the truth or lying, because they both said Carl took the money.

Therefore, only 1 of the 3 people are telling the truth. Both Alice and Beto are therefore lying, and only Carl is telling the truth.

Beto said he didn't take the money. Since we know he is lying, the culprit is Beto.

As Alice said that Carl is lying, Alice and Carl can neither tell the truth or lies together. Then we can also know that either of Carl and Beto tell truth and either of them lie; Then there are only two situations: (1)Alice:F, Carl:T, Beto:F; (2)Alice:T, Carl:F, Beto:T; in the second situation, from what Alice said we know Carl took the money and Carl said Alice didn't take the money,i.e. Carl tells truth, which is a contradiction. Then Beto lies, so he took the mony.

If what Alice said is true, then Carl took the money. But that means that Alice didn't take the money. Since any one who says the truth says the truth for both parts of their statements, then Beto is lying. Since any one who lies will lie for both parts of their statements, Beto took the money.

IMPLICATIONS TO THE THIRD DEGREE

Alice :
First degree ==> Carl's a liar. Carl's guilty.
Second degree ==> Beto is truthful. Alice's guilty.
Third degree ==> Carl's guilty. Beto's innocent.

Alice implicated both Carl (directly) and herself (by Carl's supposed lies), but this should not be the case since there's only one guilty party among them. Therefore, Alice is an innocent liar, Carl's an honest innocent while Beto's the guilty liar.

CONSIDERATION OF BOTH POSSIBILITIES

Beto : Carl took the money. I didn't take the money.

Beto accused Carl as the thief and claimed himself innocent, but if he's lying, then Beto himself is the thief while Carl's innocent. Anyway, Alice can't be the thief in both scenarios, and this just prove that Carl's being truthful while Beto's both a liar and the guilty thief.

Beto says "Carl took the money." and "I didn't take the money". If Beto is telling the truth, Carl took the money. If Beto is lying, Beto took the money. In either case, Alice didn't take the money.

Now consider Carl: he says "Beto is lying" and "Alice didn't take the money". We have already inferred that Alice didn't take the money. Thus we can conclude that Carl is telling the truth: Beto is a liar, and Beto took the money.

Alice's statement is superfluous, but consistent with Beto taking the money. Alice says Carl is a liar and Carl took the money. Both statements are lies.

假设Alice是对的，那么carl就偷了钱，第二个的回答与此不矛盾。解决此类问题的办法是假设

If Alice is telling the truth then Carl must be lying $\Rightarrow$ Alice took the money so she is lying about Carl.

If Alice is lying then Carl must be telling the truth which means Beto is lying about her not taking the money

$\Rightarrow$ Beto took the money.