Logic+ 10: UNIZOR.COM - Math+ & Problems - Logic

Notes to a video lecture on http://www.unizor.com

Logic+ 10

Problem A

Two mathematicians, Mike and David, have met their classmate Helen and asked about her birthday.
She decided to test their logical skills and answered as follows.
"My birthday is one of these days:
15th of May,
16th of May,
19th of May,
17th of June,
18th of June,
14th of July,
16th of July,
14th of August,
15th of August,
17th of August."
Then she told Mike her month and told David her day of birthday, and asked them to guess the full birthday.
After thorough thinking, Mike said: "I cannot determine the birthday, but I know for sure that David can't do it either."
David replied: "I was not able to determine Helen's birthday before you said it, but now I can."
Then, in his turn, Mike said: "Now I also know Helen's birthday."
What is the Helen's birthday?"

Solution A
Let's start with something simple.
Mike has a month of the birthday, while David has a day of the month.
Since days 18 and 19 occur only once among all the possible dates, David, if given one of these dates, would immediately realize that Helen's birthday is either 18th of June or 19th of May.
Not only he was not able to determine the birthday before Mike's first statement, Mike said that David for sure cannot determine it. How would Mike know that David has no chance to determine the birthday? Because the month given to Mike is not May nor June. Otherwise, there would be a chance that David got 18th or 19th, in which case he would be able to determine the birthday.
So, the birthday is not in May nor in June. Only if Mike has July or August he can say what he did.
From the first Mike's statement David can conclude that the month must be either July or August.
Mike, in his turn, understood that the day David was given is one of these: 14, 15, 16, 17.
The remaining possibilities are
14th of July,
16th of July,
14th of August,
15th of August,
17th of August.
David realizes that Mike went through this logic when saying that he cannot determine the birthday nor David can.
At this point David said that he knows the birthday. Hence, the day given to David cannot be the 14th, because that would leave two possibilities for the birthday - 14th of July and 14th of August. Since David said that he knows the birthday, the day given to him is 15th, 16th or 17th, as each of them is unique.
Mike also realizes that it cannot be the 14th, since David said that he knows the birthday.
Hearing that, Mike understands that it must be one of the remaining unique dates: 16th of July, 15th of August or 17th of August.
Now Mike said that he knows the birthday, which is possible only if he was given the unique month of July and, therefore, the day must be the 16th.

Answer
Based on statements Mike and David made, Helen's birthday is
the 16th of July.