Question: Peter tells Caspar and Tim: “My birthday is on one of the following days:”
- July 4,5 or 8
- August 6 or 7
- September 3 or 5
- October 3,4 or 6
Peter then tells Caspar the day of his birthday and Tim the month of his birthday. Then the following dialogue takes place:
- Caspar says: “I don’t know Peter’s birthday.”
- Tim responds: “I don’t know his birthday either, but I already knew you didn’t know.”
- Then Casper says: “Now I know his birthday!”
- And Tim adds: “Now I know it too!”
When is Peter’s birthday?
Answer: September 5.
From the first statement it follows that the day cannot be the 7th or 8th. From the second statement it follows that it has to be September or October, as otherwise Tim could not have known that Caspar did not know the answer. From the third statement it follows that the day cannot be the 3rd, as otherwise Caspar would still not be certain about the birthday. And from the last statement it follows that it has be the 5th of September, as otherwise Tim would still be in doubt if it is on the 4th or the 6th day of the month.
This puzzle was part of a Singapore Math Olympiad: Cheryl’s Birthday - Wikipedia