Knights or Knaves?

You find yourself on the Island of Knights and Knaves, a logical kingdom whose inhabitants either always lie (Knaves) or always tell the truth (Knights). You encounter three of said inhabitants, call them Alice, Bob, and Carol. You ask the following questions.

To Alice you ask, "Is Bob a Knight or a Knave?"

"A Knight," she answers.

To Bob you ask, "Are Alice and Carol both Knights or both Knaves?"

"Yes," replies Bob.

Finally, you ask Carol, "Is Bob a Knight or a Knave?"

"A Knave," she says.

Who is a Knight and who is a Knave? Click below for the answer.

Since Alice and Carol answered differently to the same question, they cannot both be Knights or Knaves, they must represent one of each type. Since this is counter to what Bob answered, he must be lying. Bob must be a Knave. That means that Carol answered truthfully, and is a Knight, and Alice did not, so she is a Knave.

True or False?

You're taking your college entrance exam, and things are not going well. You spent too much time answering the essay questions and word problems, and now you're running out of time. You turn to the final section and are relieved to find that it consists of 100 True/False questions. The bad news is that you only have 2 minutes left to complete the entire exam!

You figure it doesn't matter if you answer all True, all False, or just guess randomly. Odds are you'll probably get half of them right either way. You begin scribbling in your answers.

How many different ways (unique sequences) are there to answer 100 True/False questions? Click below for the answer.

If you're a computer programmer, you may have recognized right away that this is just a binary counting problem in disguise. Substitute 1 for True and 0 for False, and this is the same as asking how many unique values can be represented by 100 binary bits. Since that's every value from all 0s to all 1s, the answer is $2^{100}$, or 1,267,650,600,228,229,401,496,703,205,376.

King's Sentence

A man is caught poaching on the King's property. He is brought before the King to be sentenced.

The King says, "You must give me one statement. If it is true, you will killed by lions. If it is false, you will be killed by trampling of wild buffalo."

In the end, the King has to let the man go.

What was the man's statement? Click below to see the answer.

The man can get out of the punishment by making a paradoxical statement, such as "I will NOT be killed by lions." If that statement was true, the man would be killed by lions, making the statement false. Another statement that would have worked is "I will be killed by trampling of wild buffalo."

A Boy and a Girl

Two children, one boy and one girl, make the following statements.

"I am a boy" said the child with brown hair.

"I am a girl" said the child with red hair.

At least one of the children is lying. Who is the boy and who is the girl? Click below for the answer.

Both children are lying.

The child with the brown hair is the girl, and the child with the red hair is the boy.

(If only one child had lied, they would both be boys or both be girls.)