Problems

Pascal and Pierre are trying to work out Pythagoras. Being a bit obnoxious, Pythagoras puts a list of \(10\) possible dates on the table in front of them. In private, Pythag tells Pascal only the month of his birthday, and Pierre only the day of the month.

Pascal says “I don’t know Pythagoras’ birthday, but I also know that Pierre doesn’t know."
Pierre then says “Pascal was right that I didn’t know, but now I do know." Pascal replies “I know now Pythagoras’ birthday."

What is Pythagoras’ birthday?

Ignat gives two tutors a problem. He’s thinking of two different positive integers greater than \(1\), whose sum is at most \(100\). Ignat tells Pier (\(P\)) the product of the two numbers, and Sasha (\(S\)) the sum of the two numbers. Both Pier and Sasha know all of this, and then have a little conversation:

Sasha says that Pier doesn’t know the two numbers.

Pier says that he knows the two numbers.

Finally Sasha says that he knows the two numbers.
What are the two numbers?

On a bus out of the Hattius kingdom, you sit behind two wizards. The first says to the second “I’ve a positive integer number of children, all of whose ages are positive integers. The sum of their ages is the bus number, and their product is my age."

The second wizard replies “If you told me your age and how many children you had, would I be able to work out their individual ages?"

The first wizard says “No, unfortunately not." To which the second says “Aha! Now I know how old you are." What was the bus number?