Identify the Murderer

Puzzle Statement

The Police have three suspects for the murder of Mr. Cooper: Mr. Smith, Mr. Jones and Mr. Williams. Smith, Jones and Williams each declare that they did not kill Cooper.

Smith also states that Cooper was a friend of Jones and that William disliked him. Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed. William also states that he saw both Smith and Jones with Cooper the day of the killing and that either Smith or Jones must have killed him.

Can you determine who the murderer was if:

  1. One of the three men is guilty, the two innocent men are telling the truth, but the statements of the guilty man may for may not be true?
  2. Innocent men do not lie?

Identify the Murderer

Solution to Identify the Murderer

The trick to such problems is breaking them into pieces, analyzing each piece and then eliminating the inconsistent propositions.

Alternatively we can also use formal proofs by using Propositional statements and connectives. But let us not work with formal proof for now. We will do it in a simpler way this time.

Let us break down each of the suspects statement:

Jones: 

  1. I didn’t murder.
  2. I didn’t know Cooper.
  3. I was out of town on the day of murder.

Smith:

  1. I didn’t murder.
  2. Cooper and Jones were friend.
  3. William disliked Cooper.

William:

  1. I didn’t murder.
  2. Smith was with Cooper that night.
  3. Jones was with Cooper that night.

Let us evaluate all these statements with respect to the assumption that

“One of the three men is guilty, the two innocent men are telling the truth, but the statements of the guilty man may for may not be true?”

Let us evaluate Jones’ and Smith’s statement, the second statement of Jones and Cooper is contradictory. This means either of them is telling a lie and two cases arise:

  • Smith is telling a lie.
    • This means Jones and Williams must be telling truth. This means, the following statements are true:
      • Williams: Jones was with Cooper that night.
      • Jones: I was out of town that night.
    • But these are contradictory statements, hence our assumption that Smith is telling a lie is wrong.
  • Jones is telling a lie
    • This means that Smith and Williams must be telling truth.
    • We established above that Smith is telling the truth.
    • Also, Smith and William do not say any contradictory statements, hence they must be telling the truth.

 

The same argument can hold for the second assumption that “Innocent man do not lie.”

Hence, we can identify the murderer using the above arguments and he is Jones.