On my puzzle blog earlier today I set you the following puzzle:
You and your two friends Pip and Blossom are captured by an evil gang of logicians. In order to gain your freedom, the gang’s chief, Kurt, sets you this fearsome challenge. The three of you are put in adjacent cells. In each cell is a quantity of apples. Each of you can count the number of apples in your own cell, but not in anyone else’s. You are told that each cell has at least one apple, and at most nine apples, and no two cells have the same number of apples.
The rules of the challenge are as follows: The three of you will ask Kurt a single question each, which he will answer truthfully ‘Yes’ or ‘No’. Every one hears the questions and the answers. He will free you only if one of you tells him the total number of apples in all the cells.
Pip: Is the total an even number?
Blossom: Is the total a prime number?
You have five apples in your cell. What question will you ask?
Today’s puzzle has really captured your imaginations. But from reading the hundreds of comments BTL it seems many people were overthinking the answer. You need to ask a question from which one of you can deduce the total, not necessarily you.
The question you should ask is: “Is the total 15?”
Each cell has between 1 to 9 apples in them, and no two cells have the same number. So the lowest possible total is 1 + 2 + 3 = 6 apples, and the highest possible total is 7 + 8 + 9 = 24 apples.
If we eliminate all the even numbers and the prime numbers we are left with only three possible totals: 9, 15 and 21.
Case 1: the total is 15.
Kurt will answer yes to the question ‘Is the total 15?’. Since we know he is truthful, any one of you can tell him that the total is 15.
Case 2: the total is 9
Kurt will answer no to the question ‘Is the total 15?’
If the total is nine, and you have five apples. Then Pip and Blossom have four between them, so one has a single apple and the other has three. The person who has a single apple knows that the total cannot be greater than 1 + 8 + 9 = 18, and the person who has three apples knows that the total cannot be greater than 3 + 8 + 9 = 20. So, the total cannot be 21, nor is it 15, so both Pip and Blossom can deduce the total is 9.
Case 3: the total is 21
As above, Kurt will answer no to the question ‘Is the total 15?’ And following similar reasoning, Pip and Blossom have 16 apples between them. Which means one must have 9 and the other 7. And if they do, they can both deduce that the total must be more than 9, which means they know that it must be 21.
I hope you enjoyed today’s puzzle. I’ll be back in two weeks.
Meanwhile, if puzzles are your thing, you may enjoy my latest book Puzzle Ninja: Pit Your Wits against the Japanese Masters, which contains more than 200 of the most original, beautiful and interesting puzzles that have been crafted in Japan over the last few years. You don’t need to be a logician to deduce that Puzzle Ninja would make a perfect Christmas gift!
I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.