Larry, Matt, and Nick live in the strange country of Logitopia. This country is inhabited by three races of people: the type A people who always tell the truth, the type B people who always lie, and the type C people who alternately tell the truth and lie. One of the three is the president of Logitopia.
Larry makes these two statements:
- The president is of a different race from the other two.
- Matt is not the president.
Matt makes these two statements:
- The president is a type B person.
- Larry is not the president.
Nick makes these two statements:
- Exactly two of us are of the same race.
- I am not the president.
Who is the president?
Larry is the president.
Suppose Matt’s first statement is true. This would mean that he is not a type B person (who always lie), and Larry’s second statement is true. This would mean that Larry is not a type B person, and Matt’s second statement is true. Since neither Larry nor Matt is the president, Nick must be president, making both his statements false. Since Nick’s first statement is false, and since he’s a type B person while Matt is a type A person, Larry must be a type C person, making his first statement false. But since the president belongs to a different race from each of the other two, Larry’s first statement must be true. This means that Matt’s first statement is false.
Suppose Matt is president. This would mean that Larry’s second statement is false and Matt’s and Nick’s second statements are true. Then, since Matt is both the president and a type C person, and since Larry must also be a type C person if his first statement is true (which would make his first statement false), Larry’s first statement must be false, making him a type B person. Then if Nick’s first statement is true, he is a type A person, which means that all three are different races and makes Nick’s first statement false, but if his first statement is false, he is a type C person, which means that he and Matt both belong to a race that Larry does not and makes Nick’s first statement true. So Matt cannot be president.
Suppose Nick is president. This would mean that Larry’s and Matt’s second statements are both true, but Nick’s is false. Then, since Matt’s first statement and Nick’s second statement are both false, Nick’s first statement must be true. Then, since Matt and Nick are both type C people, Larry’s first statement must be false and all three must be type C people, making Nick’s first statement false. So Nick cannot be the president either, leaving only Larry.