Math Olympiad Problem

mop.txt

July 14
July 16

August 14
////August 15////
////August 17////

////May 15////
////May 16////
////June 17////
////June 18////
////May 19////


If Cheryl told Bernard 18 or 19, he would instantly know her birthday is June 18th and May 19th respectively, so she couldn't have told Bernard those days. True but this is not a valid reason to get rid of June 18 or May 19 because Albert immediately knew Bernard didn't know. See below.

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How could Albert know that Bernard doesn't know her birthday? What if she told him 18 or 19?? Answer: He knows because Cheryl DIDN'T say June or May to Albert!

Let's say Cheryl told Albert August:
If Cheryl said 15 to Bernard he wouldn't know whether it's August 15 or May 15. If Albert states that Bernard doesn't know the date, then Bernard would know that Albert knows that Cheryl didn't say May or June, and thus he would know that the only 15 is August 15 and thus her birthday. But in this case Albert still wouldn't know which day in August it is and thus we know Cheryl didn't say 15.

Doesn't work for 17 either. When Cheryl tells Bernard 17, and when Albert states that Bernard doesn't know, he would immmediately know it's August 17. But then Albert still wouldn't know what day in August it is. Albert wouldn't know if it's August 15 or August 17. So Cheryl didn't say 17 to Bernard either.

If Bernard knew immediately after Albert stated Bernard didn't know, then it must be July 16 since after elimination July 16 is the only 16 and thus the only way Bernard would know for sure. And Albert knows because Cheryl told him July.

Cheryl's birthday is July 16th.

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////There has to be another reason Albert knows Bernard doesn't know the birthday. Yes. Albert also knows that Bernard doesn't know because Cheryl didn't say August to Albert either, because if she said August then there's a chance she said 15 to Bernard (and then Bernard would immediate know it's August 15), so Albert wouldn't know for sure if Bernard knew (so he couldn't make that statement). So we can get rid of August./////

So from Albert's point of view he know's it's not May,June,or August - but Bernard doesn't know that.

Albert knows the month but not the day. For Albert the birthday can either be July 14 or July 16. If Cheryl told Bernard 14 he (Bernard) would think it's either July 14 or August 14. If Cheryl told Bernard 16 he would think it's either July 16 or May 16. Albert knows that it can't be August 14 or May 16.

Now Albert knows Bernard doesn't know because Cheryl said 16 to Bernard and Bernard doesn't know if it's May 16 or July 16. 

The fact that he said that tells Bernard that the month is July. Cheryl's birthday is July 16.