He said, she said
The boss's birthday
Alice, a manager at Big-Name-Firm Inc, thought of a riddle to give her colleagues Bryan and Claire. She told them that her birthday is on one of the following dates.
March 4, March 5, March 8 June 4, June 7 September 1, September 5 December 1, December 2, December 8Alice also told the Bryan the month of her birthday, and Claire the day. Later on, the following conversation took place.
Bryan: I don't know Alice's Birthday, but you don't know it either. Claire: I didn't know her birthday initially, but now I do. Bryan: Now I know it too.What is Alice's birthday?
The key insight in this question is tabulating the data. Given in the form of a list, it is hard to say which one of the ten dates is the solution, but things get clearer when we arrange the months as rows, and days as columns. X's mark possible solutions in the table below.
| 1 | 2 | 4 | 5 | 7 | 8 | |
|---|---|---|---|---|---|---|
| March | X | X | X | |||
| June | X | X | ||||
| September | X | X | ||||
| December | X | X | X |
We are now in good shape to analyze the first line. Imagine that the Alice's birthday was June 7 or December 2. Since there is only one X in the 2 and 7 columns, Claire would automatically be able to guess the month and deduce the solution. It follows that if Bryan was given June or December as the birthday month and no additional information, he could not be certain that Claire doesn't know the solution. We deduce that the birthday cannot be in June or December. The following table summarizes the remaining possibilities.
| 1 | 4 | 5 | 8 | |
|---|---|---|---|---|
| March | X | X | X | |
| September | X | X |
Let's look at the second line. Claire confirms she did not know the birthday originally, but she has been able to figure it out given Bryan's comment (which eliminates June and September). Looking at the updated table, Claire must have been given 1, 4, or 8, since she wouldn't be able to deduce the birtday otherwise (the 5 column contains two possibilities).
In the third line, Bryan deduces the birthday himself. The only information we could draw from Claire's previous comment was that the solution is not on the 5th day of a month. If Bryan is given March, he would still need to decide between March 4 and 8. He doesn't have any additional information, so he must have been told the birthday falls in Septebler. We conlude the solution is September 1!