Cheryl And Betty Celebrate

Let us first construct the tables for possibilities for their birthday dates

$\underbrace{\text{Cheryl's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \text{August 9} \\ \text{September 3} & \text{September 5} & \text{September 7} & \text{September 10} \\ \hline \end{array}} \qquad\underbrace{\text{Betty's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \text{August 9} \\ \text{September 3} & \text{September 5} & \text{September 7} & \text{September 10} \\ \hline \end{array}}$

We are told that Cheryl told Betty the day of the month of her birthday, and Betty told Cheryl the month of her birthday.

Cheryl then started off by claiming that Betty does not know her birthday. This tells us that day of the month written in the table for Cheryl's birthday isn't unique and thus has been written at least twice. This tells us that we can cancel out the dates whose day of the month only appears once (from Cheryl's possible dates):

$\underbrace{\text{Cheryl's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \xcancel{\text{August 9}} \\ \xcancel{\text{September 3}} & \text{September 5} & \text{September 7} & \xcancel{\text{September 10} } \\ \hline \end{array}} \qquad\underbrace{\text{Betty's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \text{August 9} \\ \text{September 3} & \text{September 5} & \text{September 7} & \text{September 10} \\ \hline \end{array}}$

However, with Betty's reply, we can only cancel out $5^\text{th}$ of September from the list of Betty's possible birthday dates as illustrated below:

$\underbrace{\text{Cheryl's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \xcancel{\text{August 9}} \\ \xcancel{\text{September 3}} & \text{September 5} & \text{September 7} & \xcancel{\text{September 10} } \\ \hline \end{array}} \qquad\underbrace{\text{Betty's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \text{August 9} \\ \text{September 3} & \xcancel{\text{September 5}} & \text{September 7} & \text{September 10} \\ \hline \end{array}}$

Afterwards, Cheryl still claims that she doesn't know Betty's birthday but she pointed out that her Birthday isn't on $7^\text{th}$ of August, updating Chery'ls possible bithday dates gives:

$\underbrace{\text{Cheryl's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \xcancel{ \text{August 7}} & \xcancel{\text{August 9}} \\ \xcancel{\text{September 3}} & \text{September 5} & \text{September 7} & \xcancel{\text{September 10} } \\ \hline \end{array}} \qquad\underbrace{\text{Betty's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \text{June 4} & \text{June 6} & \text{June 8} & \\ \text{July 4} & \text{July 5} & \text{July 6} & \text{July 8} \\ \text{August 4} & \text{August 5} & \text{August 7} & \text{August 9} \\ \text{September 3} & \xcancel{\text{September 5}} & \text{September 7} & \text{September 10} \\ \hline \end{array}}$

Following Cheryl's remark, Betty is able to determine Cheryl's birthday. This tells us that there are only one day of the month which isn't used twice, and the only possible solution to this is $\boxed{\text{Cheryl's birthday is on September 7}}$ .

Betty then made another claim that the day of the month of her birthday is a prime number. Among the number $\{3,4,5,6,7,8,9,10\}$ , only $3,5$ and $7$ are prime numbers. Thus we are able to cancel out plenty of possible dates from Betty's list:

$\underbrace{\text{Betty's possible birthday dates}}_{\begin{array} { | c c c c | } \hline \xcancel{ \text{June 4} } & \xcancel{ \text{June 6} } & \xcancel{ \text{June 8} } & \\ \xcancel{ \text{July 4} } & \text{July 5} & \xcancel{ \text{July 6} } & \xcancel{ \text{July 8} } \\ \xcancel{ \text{August 4}} & \text{August 5} & \text{August 7} & \xcancel{ \text{August 9} } \\ \text{September 3} & \xcancel{\text{September 5}} & \text{September 7} & \xcancel{ \text{September 10} } \\ \hline \end{array}}$

In the end, Cheryl is also able to figure out Betty's birthday. This tells us that there is only one month which isn't used twice, and the only possible solution to this is $\boxed{\text{Betty's birthday is on July 5}}$ .

So the total number of days between Betty's and Cheryl's birthday (both dates inclusive) is

$\text{Total number of days in July except the first 4 days of July} \\ + \\ \text{total number of days in August} \\ + \\ \text{first 7 days of September} \\ = \\ (31-4) + 31 + 7 =\boxed{65} .$

From Jul 5 to Sep 7 (both dates inclusive) we should have 65 days because July and August both have 31 days, we have 31*2 + 7-5+1 = 65