Ben: I don’t know, Mark CANNOT know either.
i.e. Ben holds a M with multiple N
Eliminate all N/6, N/12. since 6 and 12 each holds a unique value of N, as ben doesn’t hold a M with unique N, if not Mark would have known the answer to be either 7/6 or 2/12.
2)Mark: I don’t know, NOW i know.
i.e. Marks holds a N with multiple M
Eliminate all 5/M, thats left with 3 possibilities.
4/3, 8/3
1/9
And since Mark knows the answer straight from 1) to 2), he cannot be having N/3 if not he would have to have 1 or more lines of conversation to determine which of the N/3 it is.
“we know Mark can’t have any M that corresponds to the unique Ns.”
I agree it cannot be (7,6) or (2,12) because Mark would have known without knowing the month. You can eliminate June because there’s only 1 choice (4/6) left after (7,6) is canceled. (Ben would have known, if it is June)
But for Dec, if it is Dec, Ben would not have known which of the 2 remaining (1,12), (8,12) is correct. While Mark who knows the date will not know too, because there’s corresponding (1,9) and (8,3) respectively.
Let’s not use month names here, as whether you’re reading this as US or European style is confusing the issue a little bit.
All Mark is saying is that he doesn’t know the birthdate, but he can ENSURE that Ben doesn’t do know the birthdate either. The word “ensure” is key here. The only way he knows that is if Ben is given one of the unique Ns, either 7 or 2. The only way he can be absolutely 100% positive that Ben does not have a 7 or 2 is if Mark did not get a 6 or 12.
If the birthdate is x/12/1980, then Mark was given “12,” and are three possibilities for Ben’s number:
Ben has a 1 or 8, in which case Ben doesn’t know the birthdate
Ben has a 2, in which case Ben does know the birthdate
So, if the birthdate is x/12/1970, Mark cannot logically ensure that Ben does not know the birthdate.
Yes. Saying “I am sure” would have been better. So we can make two changes to the puzzle now. Add a sentence saying we are to assume both Mark and Ben are perfect logicians, and that “I can ensure” is changed to “I am sure.”
I’m assuming that the qualm is that “ensure” indicates doing something to make something certain rather than a state of mind? I’ll be honest, I didn’t even notice that until you mentioned it, and I personally use the world and hear the word used a little more generally than that. I use and hear “ensure” as a synonym for “guarantee,” in the sense of asserting something with confidence/certainty.
Should I just assume a poorly worded logic problem that would have Ben say he can ensure (or even be sure) that Mark doesn’t know the correct answer? Why can Ben be certain of that?
Mr. Tam tells Mark that N=2.
Mr. Tam “Do you know when my birthday is?”
Ben “I do not know. But I can ensure Mark doesn’t either.”
Mark: “Well. Yeah. Actually I do. It’s pretty clearly 2/12/1970. Would you mind not talking out of your ass in the future?”
Yes, if Mark or Ben lie, this doesn’t work. That’s also kind of an implicit assumption, but I guess we should put it up there with “Assume they are perfect logicians and their statements are truthful and accurate, given their state of knowledge at the time of their response.”
I get what you are saying, thanks for the clarification. Just for discussion, how about this train of thought.
“Before the Ben Answers”
0) If it has been 2/x/1970, or 7/x/1970, Mark would have known the answer from the start. Mr Tan’s question would have been brainless
“Answers”
1a) When Mark did not say anything, Ben know that it is neither of the 2 dates. Therefore, even if he knows it is xx/12/1970, He can be sure that Mark do not know which of the 2 remaining (1/12/1970 or 8/12/1970) is the truth, because there’s a corresponding (1/9/1970 and 8/3/1970) for both dates. He thus say out “his statement”
1b) Ben do not know, therefore the answer cannot be 4/6/1970, the only other xx/6/1970 choice given that 7/6/1970 is out.
Mark now knows. The only choice that he can derive the answer is 4/3/1970. I. because if it’s 1/xx/1970, 5/xx/1970 or 8/xx/1970, he wouldn’t know which month it is in. II. Since 4/6/1970 is out, the other 4/xx/1970 is 4/3/1970.
Ben follows logic 2i and he now knows the answer is 4/3/1970.
Well, not so much brainless as not being able to give us, the reader, enough to answer the question. If his birthdate were 2/x/1970 or 7/x/1970, then the conversation would go something like.
Ben: I don’t know.
Mark: I do know.
Ben: Initially I didn’t know, but now I do.
And they both do know. But we, as readers, don’t, as there are two possible answers that fit those conversations: 2/x/1970 and 7/x/1970. But with that dialogue, we’re not given enough information to come up with a unique answer, so it doesn’t work as a puzzle. (I haven’t thought about it, but maybe there is a clever way you can make the dialogue work.)
So you’re assuming Mark’s initial silence as information that Mark doesn’t know? In other words, because Ben speaks first this means Mark doesn’t know? I can see the logic of that, but I would say that it’s not a safe assumption. Maybe Ben is just the big talker in the group. Maybe Mark was still mulling over the dates and didn’t realize that he had a unique date until Ben started to talk. Maybe Mr. Tan asked them in order (although that wasn’t given in the puzzle. I guess we can add yet another clarifying point, that Mr. Tan first asks Ben the question, Do you know when is my birthday?" although I personally don’t think Mark’s silence can be taken as a definite sign of anything.)
OK, so let me double check where we’re at. This is the same question as starting with Mark saying “I don’t know.” before Ben speaks. So we’ve eliminated 7/6/1970 and 2/12/1970.
So we’re left with the same dates as in the initially explanation, but with Ben’s turn at talking:
4/3, 5/3, 8/3
4/6
1/9, 5/9
1/12, 8/12
Am I understanding you correctly so far?
So, if Ben says he doesn’t know, then he eliminates 4/6. But if he follows that up with, but he can ensure Mark doesn’t know, this eliminates 4/3, too. Ben cannot make the truthful statement that Ben doesn’t know the birthdate AND that he could be sure that Mark doesn’t know the birthdate if the birthdate is 4/3. If Mark is holding a 4, and Ben says “I don’t know the birthdate,” this means it’s not 4/6, so 4/3 becomes the only possibility. Therefore, 4/3 cannot be the birthdate either. It looks to me like 8/3 and 8/12 would be the only possibilities for that to be a truthful statement.
So if I’m continuing the logic correctly, it seems that by the end of the conversation, as readers, we’re left with both 1/12 and 5/3 as being possible answers, but with no way to disambiguate based on the information we have.
And even that doesn’t work quite right. I don’t think Ben can truthfully guarantee that Mark doesn’t know the dates at that point.
Sorry…my brain is starting to hurt. Next time, I promise to think things all the way out before I type them. But it doesn’t seem to me that your logic works, as 4/3 will have been eliminated by the statement assuring that Mark doesn’t know.
I’m not talking about any of them lying. I’m asking how Ben, not knowing what Mark was told, could possibly make the statement that he is certain Mark did not know the correct answer, before Mark even had the opportunity to answer Mr. Tau’s question! The only thing Ben can be certain of at the point he is asked is that HE does not know the answer.
Check again. He can be certain that Ben does not know the answer, since Mark was given a “9”. He can only make this statement truthfully if he was given a “3” or a “9.” The logic game is figuring out what Mark and Ben were given to make those logic statements truthful.
He can, but only if the value of M that he has been given allows it.
Mark will automatically know in the event that N = 2 (2/12) or N = 7 (7/6). Assuming perfect logic, Ben knows this. Thus, the only way that Ben can be confident that Mark can not know is if the two cases N = 2 and N = 7 are impossible. Thus M is not 6 and M is not 12. (Since, if M = 6, the case 7/6 would not be ruled out, and Ben couldn’t make the claim. Similarly for M = 12.)
First, 9 isn’t a possible answer to what Mark was given, since Mark was given the first number which would either be 1, 2, 4, 5, 7 or 8.
At the start, we don’t know what number Mark was given. We don’t know what number Ben was given. We also know that Ben and Mark do not know what number the other was given.
But what Ben DOES know is that Mark could have received the number 2 or 7 (a 20% chance of that occurring). Given this possibility, there’s no way Ben could make the statement that he can “ensure that Mark doesn’t know” the answer. Not unless he’s completely talking out of his ass.
ETA: is the logic problem written wrong? Should Mark and Ben be switched because that would make a hell of a lot more sense.