# (Advanced) Cheryl’s Birthday Puzzle

Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information:

Both numbers are integers between (including) 1 and 1000

Both numbers may also be identical.

Paul is told the product of the two numbers, Sam the sum and Dean the difference. After receiving their number, the following conversation takes place:

Paul: I do not know the two numbers.

Sam: You did not have to tell me that, I already knew that.

Paul: Then I now know the two numbers.

Sam: I also know them.

Dean: I do not know the two numbers. I can only guess one which may probably be correct but I am not sure.

Paul: I know which one you are assuming but it is incorrect.

Dean: Ok, I also know the two numbers.

What are the two numbers?

**The answer is**

**73 and 64**

Here only the final statement by Paul reveals the actual numbers.

Paul is given the product

4672

and thus cannot know what the numbers are, as possible numbers are

(146, 32), (73, 64), (584, 8) and (292, 16).

However, Sam is given the sum

137, and he knows that all possible pairs are (69, 68) … (136, 1), and if given any product of these numbers, then Paul cannot deduce the factorization as there are always more than 1, thus leading to Sam stating the “I already knew that”.

But now Paul knows that the numbers must be

73 and 64, because 137 is the only sum among 146 + 32, 73 + 64, 584 + 8 and 292 + 16, of which Sam can know 100 % certainly that Paul cannot know.

Thus Paul says that he knows what the numbers are. As Paul states this, then Sam too knows the pair is

(73, 64)

because for any other numbers Paul still could not deduce the result.

Now at this point, if we did not know what numbers were given, there are only 27 possible pairs! Of these, 13 have differences that are unique, so Dean, stating that he does not know the number, means that only the 14 remaining pairings are possible:

(4, 1), (32, 23), (32, 29), (37, 16), (41, 32), (43, 16), (53, 32), (64, 37), (64, 43), (73, 64), (89, 8), (97, 16), (101, 32) and (109, 40)

There are only certain differences, and the occurrences of numbers for these differences are as follows:

3: (4, 1), (32, 29); 9: pairs (32, 23), (41, 32), (73, 64); 21: (37, 16), (53, 32), (64, 43); 27: (43, 16), (64, 37); 69: (101, 32), (109, 40); 81: (89, 8), (97, 16)

Dean must have been given 9 as the difference, as 32 is the only number appearing twice – this is the one Dean guesses as a probable number; however, Paul knows this and states that it is not in the solution, so Dean too knows that (73, 64) is the solution.

### Your Answer

## More puzzles to try-

### What is the logic behind these ?

3 + 3 = 3 5 + 4 = 4 1 + 0 = 3 2 + 3 = 4 ...Read More »### Defective stack of coins puzzle

There are 10 stacks of 10 coins each. Each coin weights 10 gms. However, one stack of coins is defective ...Read More »### Which clock works best?

Which clock works best? The one that loses a minute a day or the one that doesn’t work at all?Read More »### Five greedy pirates and gold coin distribution Puzzle

Five puzzleFry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all ...Read More »### Magical flowers!!

A devotee goes to three temples, temple1, temple2 and temple3 one after the other. In front of each temple, there ...Read More »### Tuesday, Thursday what are other two days staring with T?

Four days are there which start with the letter ‘T‘. I can remember only two of them as “Tuesday , Thursday”. ...Read More »### How could only 3 apples left

Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »### How Many Eggs ?

A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over ...Read More »### HARD MATHS – How much faster is one train from other Puzzle

Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination ...Read More »### Most Analytical GOOGLE INTERVIEW Question Revealed

Let it be simple and as direct as possible. Interviewer : Tell me how much time (in days) and money would ...Read More »### Lateral thinking sequence Puzzle

Solve this logic sequence puzzle by the correct digit- 8080 = 6 1357 = 0 2022 = 1 1999 = ...Read More »### How did he know?

A man leaves his house in the morning to go to office and kisses his wife. In the evening on ...Read More »### Pizza Cost Math Brain Teaser

Jasmine, Thibault, and Noah were having a night out and decided to order a pizza for $10. It turned out ...Read More »### Which letter replaces the question mark

Which letter replaces the question markRead More »### Which room is safest puzzle

A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, ...Read More »### Richie’s Number System

Richie established a very strange number system. According to her claim for different combination of 0 and 2 you will ...Read More »### Srabon wanted to pass

The result of math class test came out. Fariha’s mark was an even number. Srabon got a prime!! Nabila got ...Read More »### Become Normal!!

Robi is a very serious student. On the first day of this year his seriousness for study was 1 hour. ...Read More »### Sakib Knows The Number!

Ragib: I got digits of a 2 digit number Sakib: Is it an odd? Ragib: Yes. Moreover, the sum of ...Read More »### What is the age of grand father puzzle

A boy asks his father, “what is the age of grand father?“. Father replied “He is x years old in ...Read More »