**Multi-level thinking** problems are logic puzzles in which everyone is infinitely intelligent (i.e. can reason through everything), everyone knows everyone else is reasonably intelligent, and everyone knows everyone else knows everyone else is infinitely intelligent... and so on. This is also called **perfect information**.

In these situations, the statements people make can have profound implications, even if they seem irrelevant! In this quiz we will see how seemingly innocuous statements can allow us to figure out hidden numbers in perfect information scenarios.

Alice and Bob are both given a different 1-digit number (excluding 0). They make the following statements, in order:

**Alice:** "I don't know whose number is bigger."

**Bob:** "I don't know whose number is bigger."

**Alice:** "I don't know whose number is bigger."

**Bob:** "I don't know whose number is bigger."

What is Bob's number?

The last problem shows the power of statements when considered in order. Even though Alice and Bob just repeatedly say "I don't know whose number is bigger," each statement gives more information than the last, because they have to take previous statements into account as well. Even a simple statement like "I don't know" can give information!

In general, treating each statement sequentially is the key to solving these types of logic questions, as well as keeping in mind the question "what did he know, and when did he know it?"

This question (and the next two) are all about the same puzzle, broken into three steps.

Albert and Bernard just became friends with Cheryl, and want to know when her birthday is. She gives them a list of possible dates:

May 15 | May 16 | May 19 |

June 17 | June 18 | |

July 14 | July 16 | |

August 14 | August 15 | August 17 |

Cheryl then tells Albert the **month** of her birthday (without Bernard hearing). She then tells Bernard the **day** of her birthday (without Albert hearing).

Albert:"I don't know when Cheryl's birthday is, but I know that Bernard does not know either."

Bernard:"At first I didn't know when Cheryl's birthday is, but I know now."

Albert:"Then I also know when Cheryl's birthday is."

After considering **Albert's first statement only**, which of the following months could Cheryl's birthday possibly be in?

Albert and Bernard just became friends with Cheryl, and want to know when her birthday is. She gives them a list of possible dates:

May 15 | May 16 | May 19 |

June 17 | June 18 | |

July 14 | July 16 | |

August 14 | August 15 | August 17 |

Cheryl then tells Albert and Bernard the month and day of her birthday, respectively.

Albert:"I don't know when Cheryl's birthday is, but I know that Bernard does not know either."

Bernard:"At first I didn't know when Cheryl's birthday is, but I know now."

Albert:"Then I also know when Cheryl's birthday is."

After **Bernard's statement** (and only up to that point in the puzzle), which of the following is a possible date for Cheryl's birthday?

Albert and Bernard just became friends with Cheryl, and want to know when her birthday is. She gives them a list of possible dates:

May 15 | May 16 | May 19 |

June 17 | June 18 | |

July 14 | July 16 | |

August 14 | August 15 | August 17 |

Cheryl then tells Albert and Bernard the month and day of her birthday, respectively.

Albert:"I don't know when Cheryl's birthday is, but I know that Bernard does not know either."

Bernard:"At first I didn't know when Cheryl's birthday is, but I know now."

Albert:"Then I also know when Cheryl's birthday is."

When is Cheryl's birthday?

Harry and Ron told me separately how many times they watched the movie *Titanic*. I told them "you guys both watched it, but one of you watched it once more than the other". Then they had the following conversation:

Harry: Did you watch *Titanic* more than I did?

Ron: I have no idea.

Harry: Me neither. Do you know now?

Ron: Yes, indeed!

Harry: Really? Then so do I!

What is the sum of the possible number of times Ron watched *Titanic*?

(**Hint:** Harry's initial question does not convey any information about what he knows.)

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