Maths riddles

 

Assessment started: undefined.

Item 1

In a crime scene, four people, A, B, C, D$A,B,C,D$ were involved and the following four facts were established:

1. If both A$A$ and B$B$ are guilty, then C$C$ was an accomplice.
2. If A$A$ is guilty, then at least one of B, C$B,C$ was an accomplice.
3. If C$C$ is guilty, then D$D$ was an accomplice.
4. If A$A$ is innocent then D$D$ is guilty.

Which ones are surely guilty?

• B$B$
• D$D$
• A$A$
• CSolution

  If B$B$ is innocent, then it must be C who is guilty. 

  But suppose B$B$ is guilty. Then A$A$ and B$B$ are both guilty, hence by (1)$(1)$ C$C$ is guilty too.

  This proves that if A$A$ is guilty, so is C$C$.
  Also, by (3)$(3)$, if C$C$ is guilty so is D$D$.

  Combining these two facts, we see that if A$A$ is guilty, so is D$D$.

  But by (4)$(4)$, if A$A$ is innocent, then D$D$ is guilty.

  There­fore, regardless of whether A$A$ is guilty or innocent, D$D$ must be guilty.

  So D$D$ is definitely guilty. The rest are all doubtful

  Assume  A$A$ is guilty. Then by (2)$(2)$,  either B$B$ or C$C$ is guilty.

  Solution

  Assume  A$A$ is guilty. Then by (2)$(2)$,  either B$B$ or C$C$ is guilty.

  If B$B$ is innocent, then it must be C who is guilty. 

  But suppose B$B$ is guilty. Then A$A$ and B$B$ are both guilty, hence by (1)$(1)$ C$C$ is guilty too.

  This proves that if A$A$ is guilty, so is C$C$.
  Also, by (3)$(3)$, if C$C$ is guilty so is D$D$.

  Combining these two facts, we see that if A$A$ is guilty, so is D$D$.

  But by (4)$(4)$, if A$A$ is innocent, then D$D$ is guilty.

  There­fore, regardless of whether A$A$ is guilty or innocent, D$D$ must be guilty.

  So D$D$ is definitely guilty. The rest are all doubtful