Solve the following problems using heuristic methods. Explain the process, strategies, rule or procedures used to solve the problem. 1. Based on the following facts, determine who is the murderer. a. Either Alice did not commit the murder or the weapon was a knife. b. If the weapon was a knife, then Mad Hatter committed murder. c. The statement "If the Mad Hatter did not commit the murder, then neither Alice nor the Red Queen committed the murder" is false.

Examine Statement C: Statement C is false, which means the contrapositive is true. The contrapositive states, "If either Alice or the Red Queen committed the murder, then the Mad Hatter did commit the murder." Since the contrapositive is true, and it involves a logical OR, at least one part of it must be true. This means one of the following is correct: The Mad Hatter committed the murder. The Mad Hatter did not commit the murder, but either Alice or the Red Queen did. Analyze Statement B: If the weapon was a knife, then the Mad Hatter committed the murder. Therefore, if the Mad Hatter did not commit the murder, the weapon could not have been a knife. Investigate Statement A: If Alice did not commit the murder, then the weapon was a knife. This also means that if the weapon was not a knife, Alice must be the murderer. Using heuristic reasoning: If Alice were the murderer, then by statement A, the weapon would not be a knife, which means by statement B, the Mad Hatter did not commit the murder. This fits with the true contrapositive of statement C. If the Mad Hatter were the murderer, the weapon would be a knife (statement B), which aligns with statement A but does not fit with the true contrapositive of statement C, because it implies one of the women could still be the murderer. Let's deduce: Since statement C's contrapositive must be true and it suggests the Mad Hatter could be the murderer only if Alice or the Red Queen is not, we can use statement B to see if it allows us to determine the weapon. If the Mad Hatter is the murderer, the weapon was a knife. If the weapon was a knife, Alice did not commit the murder (from statement A). But since statement C’s contrapositive is true and we've established that the Mad Hatter being the murderer means Alice did not do it, the Red Queen cannot be the murderer either because that would make statement C's contrapositive false. Therefore, by elimination: The Mad Hatter must be the murderer since this is the only scenario that doesn't contradict any statements. The weapon was a knife, satisfying both statements A and B without contradicting the true contrapositive of statement C. Hence, the Mad Hatter is the murderer.