Deducing a birthday from month and day clues
Question
A birthday is one of these ten dates: March 4, March 5, March 8, June 4, June 7, September 1, September 5, December 1, December 2, December 8. A is told only the month and B only the day. A says: "I do not know the date, and I know B does not know it either." B replies: "I did not know before, but now I do." A then says: "Now I know too." What is the birthday?
Short answer
September 1. Days 7 and 2 are unique, so A's month cannot be June or December. That leaves March 4, 5, 8 and September 1, 5; B now knows, so the day is not 5, leaving March 4, March 8 and September 1. A then knows, so the month has a single remaining date: September.
Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 2.2 Logic Reasoning, 'Birthday problem (logic)', p.4; question and answer rewritten by SelfMadeQuant.com.
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