Sunday night quiz? Are you S.M.R.T.?
Okay kids, play at home with Vern and I!
Vern’s been working at this all day, and just put it in my lap. I’m still thinking on it, but I’ve only had a few minutes to ponder it. Perhaps you know the answer?
A teacher wants his students to guess his birthday.
He gives student A the month and student B the day, but forbids them to tell anyone. Now he gives them these possibilities:
March 4, March 5, March 8,
June 4, June 7,
September 1, September 5,
December 1, December 2 and December 8.
A says: If I don’t know, he doesn’t know either.
B says: First, I didn’t know, but now I know.
A says: Well, if he knows, then I know it too.
What’s the teacher’s birthday?
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The original puzzle had names like Dr Strange, so I was on the path of the riddle being metaphorical… and possibly all about quantum physics and quarks and things.
It is not.
This is pure logic… and… there is a solution. I vaguely remember proofs in mathematics… but, wasnt able to apply it. That was a long time ago mind you.
It would have to be June 7. Right?
Russell: Nope, can’t be — because if it was, then Student B would have known right away. He only knew AFTER the Student A spoke…
A would reason that B would know if it were June 7 or December 2, because the numbers of those dates are not repeated.
A’s first statement may be reworded as “if he knows, then I know” by some principle of logic whose name I can’t recall and couldn’t be bothered to Google for right now. That’s true because B could only know if it were June 7 or December 2, and if B knew, then A could determine which of those two options it were, because A knows the correct month.
Given that A said that, B would reason that the month was either June or December. B has therefore reduced the possible options to June 4, June 7, December 1, December 2, and December 8.
But B said he didn’t know at first, so it can’t be June 7 or December 2 for the reasons given above. Therefore, B has reduced the candidates to June 4, December 1, and December 8. Of course there are no repeated numbers in these candidates, so B now knows the date, whatever it is, and announces it.
A would have reasoned all of the above. If A knew the date was in December, then A would be unable to say which date it was. But A is claiming they do know. That means A’s knowledge must be that the birthday is in June.
Therefore, the birthday is June 4.
A winner is random anonymous lurker?!
Oops. Erase “and announces it” from paragraph 4. The logic still holds, I think.
OMG…. Wow! I have a client emergency to deal with…. but, this looks damned good!
Yeah! Love it! Brilliant! Thank you for putting me out of my misery on this.
Who are you?
Another anonymous lurker pipes in…
You cannot reword A’s first statement as “if he knows, then I knowâ€. You can, however, reword it to say, “I don’t know, and I know B doesn’t know either”. The first part of this statement (“I don’t know”) is clear — there is no way A can know since there are multiple options for every month listed. The second part of the statement (“I know B doesn’t know either”) can only be made if the month is not June or December. If the teacher’s birthday was in one of those months, it is possible that B would know the answer since it contains unique days (7 and 2, respectively). A knows that B cannot know because A knows the month is March or September.
Once B realizes this, he can figure out the teacher’s birthday because the day he was given only appears once in the remaining options. Therefore, you can eliminate March 5 and September 5.
When B makes this known, A can figure out the teacher’s birthday because the month he was given only appears once in the remaining options. You can eliminate March 4 and March 8, leaving you with September 1.
Theresa:
We can’t reword the first statement as you suggest. I can’t prove that without resorting to truth tables though, and it’s inconvenient to do that in this comment.
Even if we could reword the first statement as you suggest, it wouldn’t be true: When A speaks, they don’t know whether or not B knows, because for all A knows, B might have been told the date was 2 or 7, in which case B would know it was December 2 or June 7, respectively.
Finally, we can so reword the first statement as I originally suggested, by the principle of Modus Tollens. This time I went and looked it up:
http://en.wikipedia.org/wiki/Modus_tollens
No, wait, I’m wrong. I mean I’m still right, but it’s not Modus Tollens. It’s transposition:
http://en.wikipedia.org/wiki/Transposition_%28logic%29
Argh.
The thing is, 23laughingDragons, you cannot use propositional logic to solve this puzzle. In propositional logic, a proposition is a statement that is either true or false all the time. That is, we can assume that if the same proposition is used more than once, all occurrences have the same meaning.
In this case, a statement like “X knows the answer” does not satisfy that definition. Such a statement can be false at one point and true some time later. (This is the whole point of the riddle. At the beginning, nobody knows the answer and at the end, both A and B know it).
With regard to the re-wording I originally suggested, let me explain further…
When A says, “If I don’t know, he doesn’t know either”, it is clear that A cannot know since he was only given the month, and multiple birthday options are provided for every month listed. If we assume all the characters are telling the truth, we can then deduce that B doesn’t know either (“If A doesn’t know, B doesn’t know either”).
But, how can A be sure of this? In your example you suggest that “B might have been told the date was 2 or 7, in which case B would know it was December 2 or June 7, respectively”. However, if B had been told the date was 2 or 7, then A must have been told the month was December or June. The only way A can be sure that B does not know the answer at the beginning of the dialogue is because he knows the month is not December or June.
Make sense?
Jeez, both of you make a lot of sense. I’m guessing you are both university grads majoring in mathmatics? I’m excellent at pattern recognition, and algebraic theory, but obviously reasoning isn’t my forte.
A winner is Theresa!
I think you’re right after all. It makes perfect sense to me as long as the first statement is
“I know that he doesn’t know.”
instead of the more confusing
“If I don’t know, he doesn’t know either.”
I really want to put student A in detention for the original wording. It just doesn’t make any sense unless A is purposefully trying to confuse. Extreme detention.
Oh man, this feels horrible. I bet this is worse than how Vern felt. Arrrrrrgh. Someone tell him.
Hooray! I will now happily return to my lurker status.