CHERYL'S BIRTHDAY - an irrefutable solution
SINGAPORE'S SEEMINGLY MIND-BOGGLING MATHEMATICS PROBLEMApparently, there is a big fuss being made about the above maths problem which was originally a question from the Singapore and Asian Schools Math Olympiad (SASMO) contest meant for Secondary 3 and 4 students. People are questioning why such a question is now being set as homework for Primary 5 students.
Well, I'm more interested in the maths problem itself.
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Official 'July 16' Solution
I think there is something problematic about this official solution. Its basic premise is that the key to the solution lies in the answer to the question: "Why does Albert know that Bernard does not know Cheryl's birthday?" It went on to assume that the reason is that Albert must have been told that Cheryl's birthday month is either July or August. That's because this would have prevented the possibility of Bernard being told that her birth day is 18 or 19. If indeed Bernard was told that her birth day is 18 or 19, he would have known straightaway that her birthday is on June 18 or May 19 respectively.
The grave error here is to mistake possibility for certainty. This is only one possibility of how Albert knew that Bernard did not know Cheryl's birthday at the outset. There are several other possibilities: Albert must have known that Cheryl is not so stupid as to render the guessing game pointless by giving the game away right at the beginning in telling Bernard that her birth day is 18 or 19; Albert knew that if Bernard knew her birthday at the beginning he would have blurted out the answer without waiting for Albert to begin the conversation since finding out her birthday was their original intention anyway; Albert might be merely making a statement of mutual ignorance in the hope of getting some clues from Bernard's response; etc, etc.......
So, it is reckless and presumptuous to rule out all the May and June dates just based on one possibility. In a logical analysis, all possibilities must be explored and excluded. What if that one possibility fails to materialize, the whole argument collapses. It could well be that even if Albert was told that her birth month is May or June, he might still say that he knew Bernard did not know her birthday at the outset. So, I think that too much weight is placed on Albert's initial statement that he knew that Bernard did not know. How Albert knew that, is still an open question as Albert did not tell us how he came to that conclusion; and it is against logic to prefer one possibility over all others without any factual confirmation.
Furthermore, for the official solution to work, Bernard must also think the same way as Albert. This does not make sense. From his point of view, he did not know Cheryl's birthday because Cheryl's birth day as told to him was not 18 or 19. To him, he would not need to rule out May and June completely to know that he did not know Cheryl's birthday at the outset. Whether Bernard presumed that Albert's declaration of the former's ignorance was because Albert had been told that her birth is July or August is another open question.
So, this official solution is unlikely, unrealistic, twisted, complicated, tentative and rather inelegant.
Contrast that with this 'August 17' solution:
All three of them knew that Cheryl's birthday is never going to be May 19 or June 18, or else it was pointless to play this game. Albert first declared that he did not know Cheryl's birthday and claimed that he knew Bernard didn't know too.
Bernard did not deny that, but deduced that May 19, June 18 and June 17 can all be ruled out. So, the remaining possibilities were May 15, May 16, July 14, July 16, August 14, August 15 and August 17. If the number told to him was 14, it could be July or August; if 15, then May or August; if 16, then May or July; and if 17, then there is no ambiguity ... BINGO! The answer must be August 17.
Bernard promptly stated that despite his initial ignorance, Albert's corresponding declaration of his mutual ignorance had helped him come to an undeniable conclusion. On hearing that, Albert knew that the answer must be the odd unpaired number 17, that is August 17.
This solution is based on common sense, factual statements and logical deduction. It uses no presumptions and shows no preference for one possibility over another. It does not require Albert nor Bernard to read each others' minds. It is simple and elegant and I'm sure Ockham will approve. Not to mention, Cheryl!
Postscript
Which of these 2 solutions is 'correct'?
As a mathematical problem, the official solution is impeccably correct, but as a philosophical or logical problem, the August 17 solution is more convincing and passes the test of Ockham's Razor better.
Which solution you prefer will depend on whether you think it to be a mathematical problem disguised as a logical puzzle or is it the other way round, a philosophical problem to be analyzed in abstract terms. Do we follow Aristotle's direction to use reason to judge earthly matters or do we direct our gaze heavenwards like Plato?
**For those who have enjoyed this article, please read my follow-up at the link below:
CHERYL'S BIRTHDAY - The Final Verdict
Read at:
http://singaporedialectic.blogspot.sg/2015/04/cheryls-birthday-it-must-be-sign-of.html?m=1
I think there is something problematic about this official solution. Its basic premise is that the key to the solution lies in the answer to the question: "Why does Albert know that Bernard does not know Cheryl's birthday?" It went on to assume that the reason is that Albert must have been told that Cheryl's birthday month is either July or August. That's because this would have prevented the possibility of Bernard being told that her birth day is 18 or 19. If indeed Bernard was told that her birth day is 18 or 19, he would have known straightaway that her birthday is on June 18 or May 19 respectively.
The grave error here is to mistake possibility for certainty. This is only one possibility of how Albert knew that Bernard did not know Cheryl's birthday at the outset. There are several other possibilities: Albert must have known that Cheryl is not so stupid as to render the guessing game pointless by giving the game away right at the beginning in telling Bernard that her birth day is 18 or 19; Albert knew that if Bernard knew her birthday at the beginning he would have blurted out the answer without waiting for Albert to begin the conversation since finding out her birthday was their original intention anyway; Albert might be merely making a statement of mutual ignorance in the hope of getting some clues from Bernard's response; etc, etc.......
So, it is reckless and presumptuous to rule out all the May and June dates just based on one possibility. In a logical analysis, all possibilities must be explored and excluded. What if that one possibility fails to materialize, the whole argument collapses. It could well be that even if Albert was told that her birth month is May or June, he might still say that he knew Bernard did not know her birthday at the outset. So, I think that too much weight is placed on Albert's initial statement that he knew that Bernard did not know. How Albert knew that, is still an open question as Albert did not tell us how he came to that conclusion; and it is against logic to prefer one possibility over all others without any factual confirmation.
Furthermore, for the official solution to work, Bernard must also think the same way as Albert. This does not make sense. From his point of view, he did not know Cheryl's birthday because Cheryl's birth day as told to him was not 18 or 19. To him, he would not need to rule out May and June completely to know that he did not know Cheryl's birthday at the outset. Whether Bernard presumed that Albert's declaration of the former's ignorance was because Albert had been told that her birth is July or August is another open question.
So, this official solution is unlikely, unrealistic, twisted, complicated, tentative and rather inelegant.
Contrast that with this 'August 17' solution:
All three of them knew that Cheryl's birthday is never going to be May 19 or June 18, or else it was pointless to play this game. Albert first declared that he did not know Cheryl's birthday and claimed that he knew Bernard didn't know too.
Bernard did not deny that, but deduced that May 19, June 18 and June 17 can all be ruled out. So, the remaining possibilities were May 15, May 16, July 14, July 16, August 14, August 15 and August 17. If the number told to him was 14, it could be July or August; if 15, then May or August; if 16, then May or July; and if 17, then there is no ambiguity ... BINGO! The answer must be August 17.
Bernard promptly stated that despite his initial ignorance, Albert's corresponding declaration of his mutual ignorance had helped him come to an undeniable conclusion. On hearing that, Albert knew that the answer must be the odd unpaired number 17, that is August 17.
This solution is based on common sense, factual statements and logical deduction. It uses no presumptions and shows no preference for one possibility over another. It does not require Albert nor Bernard to read each others' minds. It is simple and elegant and I'm sure Ockham will approve. Not to mention, Cheryl!
Postscript
Which of these 2 solutions is 'correct'?
As a mathematical problem, the official solution is impeccably correct, but as a philosophical or logical problem, the August 17 solution is more convincing and passes the test of Ockham's Razor better.
Which solution you prefer will depend on whether you think it to be a mathematical problem disguised as a logical puzzle or is it the other way round, a philosophical problem to be analyzed in abstract terms. Do we follow Aristotle's direction to use reason to judge earthly matters or do we direct our gaze heavenwards like Plato?
**For those who have enjoyed this article, please read my follow-up at the link below:
CHERYL'S BIRTHDAY - The Final Verdict
Read at:
http://singaporedialectic.blogspot.sg/2015/04/cheryls-birthday-it-must-be-sign-of.html?m=1
