(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-
Boxing Match Riddle
Usually, a boxing match consists of twelve rounds. In a particular friendly match between two heavyweights, the match was finished ...Read More »Number arrangement
Arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 above and below the division line in ...Read More »Once he could see
It does not have eyes, but once it did see. Once it had thoughts, but now it is white and ...Read More »Spot The Whale
High-IQ individuals often excel in such optical illusion challenges, showcasing their ability to process complex visual information rapidly and accurately. ...Read More »Two Letters name
My name is two letters. You call many people by this name, but yet their first name is not this. ...Read More »A game of 2014 cards
There is a table with a row of 2014 cards. Each card has a red side and a blue side. ...Read More »Easy to in Hard to out
As easy as you can get into me, it is as hard to get out of me. Do you know ...Read More »How will you Market your product?
How will you sell a Refrigerator to an Eskimo?Read More »Can you find out the result of the individual games?
Aaron, Brad, Christopher, Danny and Elvis decided to play a game of tiddly winks. In this game they decided that ...Read More »Same father for women and daughter puzzle
A women and her daughter walked into a restaurant. A man walked past and the women both said, “Hello, Father”. ...Read More »Whats Next?
Fill in the next line: 9 7 5 9 4 9 6 3 4 5 3 6 1 8 2 ...Read More »Finding gloves in dark riddle
A lady has some fine gloves and hats in her closet 14 blue,25 red and 45 yellow. The lights are ...Read More »You have a glass of water
You have a glass of water that looks about half full. How can you tell, only using the glass of ...Read More »The Alert Consumer
An alert consumer noticed that both of these labels for identical food products have errors. Can you find the ...Read More »Hard boiling 20 eggs puzzle
If it takes twenty minutes to hard-boil one goose egg, How long will it take to hard-boil four goose eggs?Read More »Which car goes into which road?
Which car goes into which road?Read More »What number is X?
What number is X? 16 06 68 88 X 98Read More »Always in demand
Most ask for it yet some hate to face it. For some it is pain and for others it is ...Read More »What two things can you never eat?
What two things can you never eat for breakfast?Read More »Floors Puzzle
We live in a large tower block, 10 floors high. My 5 year old son goes to school each day ...Read More »