A puzzle by National Security Agency mathematician Stephen C., from the agency’s July 2015 Puzzle Periodical:
Charlie presents a list of 14 possible dates for his birthday to Albert, Bernard, and Cheryl.
- Apr 14, 1999
- Feb 19, 2000
- Mar 14, 2000
- Mar 15, 2000
- Apr 16, 2000
- Apr 15, 2000
- Feb 15, 2001
- Mar 15, 2001
- Apr 14, 2001
- Apr 16, 2001
- May 14, 2001
- May 16, 2001
- May 17, 2001
- Feb 17, 2002
He then announces that he is going to tell Albert the month, Bernard the day, and Cheryl the year.
After he tells them, Albert says, “I don’t know Charlie’s birthday, but neither does Bernard.”
Bernard then says, “That is true, but Cheryl also does not know Charlie’s birthday.”
Cheryl says, “Yes, and Albert still has not figured out Charlie’s birthday.”
Bernard then replies, “Well, now I know his birthday.”
At this point, Albert says, “Yes, we all know it now.”
What is Charlie’s birthday?
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When Albert claims that Bernard does not know Charlie’s birthday, he is saying that he knows that the correct day occurs more than once in the list. In other words, he is saying that Charlie’s birthday is not Feb 19, 2000, and the only way he could know that is that he knows that the month is not February.
So by making this claim Albert has reduced the list for everyone to:
- Apr 14, 1999
- Mar 14, 2000
- Mar 15, 2000
- Apr 16, 2000
- Apr 15, 2000
- Mar 15, 2001
- Apr 14, 2001
- Apr 16, 2001
- May 14, 2001
- May 16, 2001
- May 17, 2001
When Bernard says that it is true that he does not know Charlie’s birthday, it tells everyone that even with the restricted list the correct day occurs more than once in the list. So everyone can thus eliminate May 17, 2001.
Furthermore, the claim that Cheryl also does not know Charlie’s birthday is a claim that Bernard knows that the year occurs more than once on the remaining list.
This rules out Apr 14, 1999, and since Bernard could only rule this out by knowing that the day was not 14, everyone can further reduce the list to:
- Mar 15, 2000
- Apr 16, 2000
- Apr 15, 2000
- Mar 15, 2001
- Apr 16, 2001
- May 16, 2001
When Cheryl says that Albert still has not figured out Charlie’s birthday, she is telling everyone that given the new list the month occurs more than once and thus rules out May 16, 2001. This tells everyone that Cheryl knows that the year is not 2001 and the list can be reduced to:
- Mar 15, 2000
- Apr 16, 2000
- Apr 15, 2000
When Bernard then claims to know the date he is saying that the day occurs only once among the 3 remaining choices, thus telling everyone that Charlie’s birthday is Apr 16, 2000.
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