$\textbf{Birthday}$
John and Mary know that their boss Joe's birthday is one of the following $10$ dates: Mar $4$, Mar $5$, Mar $8$, Jun $4$, Jun $7$, Sep $1$, Sep $5$, Dec $1$, Dec $2$, and Dec $8$.
Joe tells John only the month of his birthday, and tells Mary only the day.
After that, John first says: “I do not know Joe’s birthday, Mary doesn’t know it either.”
After having heard what John said, Mary says "I did not know Joe's birthday, but I know it now."
After having heard what Mary said, John says "I know Joe's birthday now as well."
What is Joe's birthday?
$\textbf{Answer}$
September $1$.
$\textbf{Analysis}$
This is a logic problem. The first step to solve a logic problem is to find an intuitive and visual way to describe all the given conditions and intermediate results. Otherwise, it will be hard to track all these information. For this problem, a table like the one below is helpful. The clear grids are all the possible dates.
For the birthday to be uniquely determined, it must be the only one possible date in one row (for John to know) or in one column (for Mary to know). The fact that John knows for sure Mary cannot tell means that Joe's birthday cannot be June $7$ or Dec $2$. In another word, John knows that Joe's birthday is in neither June nor Dec. Let's shade these two rows now.
Now, if Mary knows exactly Joe's birthday, then the date must be one of Sep $1$, Mar $4$, and Mar $8$ because these days are unique on their respective columns. This means that the date cannot be $5$. Let's shade this column now.
Now, for John to be sure, the date has to be Sep $1$.