Sum and Product Puzzle

X and Y are two different integers, greater than 1, with sum less than or equal to 100. S and P are two mathematicians; S knows the sum X+Y, P knows the product X*Y, and both are perfect logicians. Both S and P know the information in these two sentences. The following conversation occurs:

S says "P does not know X and Y."
P says "Now I know X and Y."
S says "Now I also know X and Y!"

What are X and Y?

Singapore Birthday Problem : When is Cheryl's Birthday ?

Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.

Cheryl thought a second and said,
“I’m not going to tell you, but I’ll give you some clues.”
She wrote down a list of 10 dates:

May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17

“My birthday is one of these,” she said.
Then Cheryl whispered in Albert’s ear the month and only the month of her birthday.

To Bernard, she whispered the day, and only the day of her birthday.

“Can you figure it out now?” she asked Albert.

Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.
Bernard: I didn’t know originally, but now I do.
Albert: Well, now I know, too!

When is Cheryl’s birthday?

ANSWER!

Lets first analyze all the given birth dates. The dates can be easily viewed as

Now will analyze the conversation in detail

Line 1) Albert Says: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.
All Albert knows is the month, and every month Cheryl mentioned has more than one possible date, so of course he doesn’t know when her birthday is.

The only way that Bernard could know the date with a single number, however, would be if Cheryl had told him 18 or 19, since of the ten date options only these numbers appear once, as May 19 and June 18.

For Albert to know that Bernard does not know, Albert must therefore have been told July or August (not June or May), since this rules out Bernard being told 18 or 19.

Thus, only possible months for Albert are : July and August
And, only possible days for Bernard are : 14, 15, 16 and 17

Thus, the solution space is now reduced to :

Line 2) Bernard: At first I don’t know when Cheryl’s birthday is, but now I know.

Bernard has deduced that Albert has either August or July. If he knows the full date, he must have been told 15, 16 or 17, since if he had been told 14 he would be none the wiser about whether the month was August or July. Each of 15, 16 and 17 only refers to one specific month, but 14 could be either month.

Thus, the solution space is now reduced to :

Line 3) Albert: Then I also know when Cheryl’s birthday is.

When Albert says that he also knows the answer, he doesn't have the Bernard's data still ! He doesn't know that the day with Bernard is 15, 16 or 17. Then, how did he came to conclusion that he also knows the answer. That means he has July as month as there is only one day associated with July. So, he is sure that now I also know the answer.

The answer, therefore is July 16.

5, 15, 1115, 3115, 132115, 1113122115, 311311222115, ?

What is the next number in this sequence:

5, 15, 1115, 3115, 132115, 1113122115, 311311222115, ?

True and False Statements

Which of the following statements are true and which are false?
1. Only one of the statements is false.
2. Exactly two of the statements are false.
3. Only three of the statements are false.
4. Exactly four of the statements are false.
5. All five of these statements are false.