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Birthday Problem

Problem​

You and your colleagues know that your boss A's birthday is one of the following 10 dates:

  • Mar 4, Mar 5, Mar 8
  • Jun 4, Jun 7
  • Sep 1, Sep 5
  • Dec 1, Dec 2, Dec 8

A told you only the month of his birthday, and told your colleague C only the day. After that, you first said: "I don't know A's birthday; C doesn't know it either." After hearing what you said, C replied: "I didn't know A's birthday, but now I know it." You smiled and said: "Now I know it, too." After looking at the 10 dates and hearing your comments, your administrative assistant wrote down A's birthday without asking any questions. So what did the assistant write?

Hints​

Hint 1

There are 3 mentioned dialogs that eliminate the birthday candidates in sequence.

Hint 2
  • You and C do not know the birthday
  • You say that you and C cannot figure out the birtday. From this clue, you can start eliminating candidates
  • C says now C knows it. Start eliminating more candidates
  • Now you know it. Come up to answer

Solution​

Show Solution

Sep 1

Show Explanation

Explanation:

First, remember:

  • You know the month.
  • C knows the day.

There are some days (like 2 and 7) that only appear once, so if C had gotten those days, he would know the birthday right away. But you say C doesn’t know it, so the day cannot be 2 or 7. That means the birthday is not in June or December, because those months have days 2 or 7.

Now the possible months left are March and September.

Next, C hears this, and says “now I know the birthday.” That means C’s day must be unique among the March and September dates:

  • March days: 4, 5, 8
  • September days: 1, 5

Day 5 appears in both March and September, so if C had 5, he still wouldn’t know. So the birthday cannot be March 5 or September 5.

That leaves:

  • March 4, March 8, September 1

Finally, you say “now I know it, too.” If the month had been March, you still wouldn’t know because March has two dates left (March 4 and March 8). But you do know — so the month must be September, with only one date left: September 1.