Jack and Jill are students of Mr. Hill. Mr. Hill's birthday is on $M$ month, $D$ day. Neither student knows when Mr. Hill's birthday is, but both know it is one of the dates below. Mr. Hill told Jack the $M$ value and Jill the $D$ value. Then Mr. Hill asked both of them when is his birthday.
Mr. Hill's birthday is one of the days below (format: month/day):
3/4 3/5 3/8 6/4 6/7 9/1 9/5 12/1 12/2 12/8
- Jack: I don't know, and Jill surely doesn't know either.
- Jill: At first I didn't know either, but now I do.
- Jack: Oh, then I know it too.
Based on the above conversation between Jack and Jill, determine Mr. Hill's birthday. Give your step by step reasoning.
Answer
The answer is
9/1
Day 2 only shows up once (12/2), and day 7 only shows up once (6/7). Days 1, 4, 5, 8 each show up at least twice. Jack says that Jill surely doesn't know, and therefore neither day 2 nor day 7 can be combined with Jack's month M. This means that $M \ne 12$ and $M \ne 6$.
After Jack's first statement, the list of dates has been reduced to 3/4 3/5 3/8 9/1 9/5.
Jill (knowing D) can deduce M from the new list. This means $D \ne5$, but the other values 1,4,8 are still possible. The list reduces to 3/4 3/8 9/1.
Jack (knowing M) can deduce D from the new list. This means $M \ne 3$ and onlly leaves one possible answer stated above.
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