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This problem has been making the round on the internet. The solution provided gives one answer, and I don't disagree with the logic to arrive at that answer. However, it seems to me that there is at least one additional solution.

I will post the problem before the given solution or mine so as not to ruin it for you if you want to attempt yourself first:

enter image description here

SOLUTION FOLLOWS

The official solution says the birthday is July 16th. As I said, I don't disagree with the logic, however it seems to me that Aug 17th is also a viable solution. Here's my thinking:

If B was initially told the birthday was on the 17th he would not know the correct date. A, having been told the month was August, would know that B couldn't possible know the correct date, but would also have to admit that he didn't know. However, by admitting that he doesn't know the birthday, B would immediately know that Aug 17th was correct. This is because if June was the correct month, A would have known the birthday was June 17 (since A knows B doesn't know, and there is only one possible 18 date, A would know it was June 17). Since A doesn't know, B can rule out June 17th, and now knows it must be the correct date. A, following the same logic, could then come to the same conclusion after B announces he knows.

Thoughts?

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    $\begingroup$ See: mark.github.io/olympiad , as posted on reddit.com/r/math by McPhage $\endgroup$
    – Krishan Bhalla
    Apr 15, 2015 at 22:07
  • $\begingroup$ That was an excellent link- thanks for sharing $\endgroup$
    – Mike Andrejkovics
    Apr 15, 2015 at 22:26
  • $\begingroup$ You may also enjoy googling "wise men puzzle". $\endgroup$
    – Nameless One
    Apr 16, 2015 at 0:45

1 Answer 1

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Suppose the birthday were August 17, as you say. Then A is told "August" and B is told "17".

A's first statement is correct; he does not know the date, and all of (aug.14), (aug.15), (aug.17) would give B one of (14,15,17), all of which leave B ignorant of the date. This statement reduces the month to July or August.

B's statement is correct; knowing the month is July or August, he now knows it cannot be June 17, so must be Aug. 17.

However now A's second statement is incorrect. He does not know the birthday, because B might have been told 17 or 15; in either case he would have made the same statement. Had B been told 15, he would have known August 15, since it cannot be May 15.

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  • $\begingroup$ Yes- I totally understand this reasoning, but here's my question: $\endgroup$
    – Mike Andrejkovics
    Apr 15, 2015 at 22:15
  • $\begingroup$ When B announces he now knows the answer, A could reason as follows: How did B determine the date simply from me saying that I didn't know? It must be that he was able to eliminate one of two possible choices s a result of my statement. If the answer was June 17, then he would have expected me to know the answer after realizing that he didn't know. Since I did NOT know the answer, he could then reason that June 17th is incorrect (having been given 17 as the date) and conclude that Aug 17 is correct. $\endgroup$
    – Mike Andrejkovics
    Apr 15, 2015 at 22:21
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    $\begingroup$ A's first statement is equivalent to "The month is July or August". B's statement is equivalent to "Knowing the month is July or August, plus the number, is enough to get the date uniquely". This would be true for both August 17 and August 15, so A can't tell which it is. $\endgroup$
    – vadim123
    Apr 15, 2015 at 22:38
  • $\begingroup$ ^^^That was exactly what I needed. I understand where my error was now. $\endgroup$
    – Mike Andrejkovics
    Apr 15, 2015 at 22:42

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