(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
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
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.
More puzzles to try-
- You have four 9’s and you may use any of the (+, -, /, *,.) as many times as you like. ...Read More »
- Take away the first letter still sound the same. Take away the last letter still sound the same. Even take ...Read More »
- In the city of Mexico, the following facts are true: No two inhabitants have exactly the same number of hairs. ...Read More »
- What does this rebus picture means ?Read More »
- All the rivers are magical, moment a worshiper crosses a river with any number of flowers, it becomes double. (For ...Read More »
- There are two tribes in Mars, Lie tribe and Truth Tribe. Lie tribe always speaks lie, True tribe always speaks truth. You meet ...Read More »
- Jacob buys a horse for $80, then sells it for $90. He buys the horse back for $120, and then ...Read More »
- How many points are there on the globe where, by walking one mile south, then one mile east and then ...Read More »
- There is a box in which distinct numbered balls have been kept. You have to pick two balls randomly from ...Read More »
- Our enemy challenges you to play Russian Roulette with a 6-cylinder pistol (meaning it has room for 6 bullets). He ...Read More »
- What spends the day at the window, goes to the table for meals and hides at night?Read More »
- What loses its head in the morning and gets it back at night?Read More »
- What is the Surname of Barbie Doll?Read More »
- You are driving a car from New York to Washington DC. Car no is DZ-2817. The distance is 226.0 miles. ...Read More »
- Sally and three of her friends decided to plant a new tree in their yard to celebrate Go Green Day. The new ...Read More »
- A fisherman decided to become a golfer. In order to buy his membership he had to sell his boat. How ...Read More »
- Two unemployed young men decided to start a business together. They pooled in their savings, which came to Rs 2,000. ...Read More »
- Assume 9 is twice 5; how will you write 6 times 5 in the same system of notation?Read More »
- What two whole, positive numbers that have a one-digit answer when multiplied and a two-digit answer when added?Read More »