Update: Several readers spotted a mistake in the setting of the puzzle. Apologies. (Prem, who set the puzzle, responds below the line). This is the first time in more than three years that I’ve set a puzzle with such an issue. In order to stop this happening again, if anyone would like to be a ‘puzzle-tester’ for this blog please get in touch with me on the email address below.
In my puzzle blog earlier today, I set you the following “common knowledge” puzzle:
Nine balls - marked 1, 2, 3, 4, 5, 6, 7, 8 and 9 - are in a lottery machine. The machine dispenses one ball each to three people, Apple, Bean and Cherry. Each person knows only the number of their own ball; they do not know the balls that the others were given, nor the ones left in the machine.
Before the game begins, each of them show their balls to a fourth person, Zog, who says: “On one of the balls is a number that is the sum of the numbers on the other two balls.”
At which point the following bantz begins:
- Apple: “There are 8 possibilities for Bean’s ball.”
- Bean: “There are 8 possibilities for Cherry’s ball.”
- Cherry: “There a are 4 possibilities for Apple’s ball.”
- Apple: “I know Bean’s ball!”
- Bean: “I know Cherry’s ball!”
What is Cherry’s ball?
We can assume that the laws of puzzleland apply, which is that all participants have PhDs in logic and they are truthful at all times.
Solution
The first challenge here is to figure out why Zog’s statement is relevant.
If the sum of two of the balls is equal to the third, then we can deduce that it is impossible for both the 8 and the 4 to be among the three selected balls. This is because if the 8 and the 4 were selected, the third ball would have to also be 4, so 4 + 4 = 8, but this is impossible since there is only one 4-ball.
So, if Apple had either the 8, she would know that Bean does not have the 4, and if she had the 4, she would know Bean does not have the 8. In either case she would deduce that there are only seven possibilities for Bean’s ball, i.e it could be any ball exception the 8-ball and the 4-ball.
In the question, Apple says there are 8 possibilities for Bean’s ball, we know she does not have either the 4 or the 8.
For the same reasons, when Bean says there are 8 possibilities for Cherry’s ball, we can deduce he doesn’t have the 4 or the 8 either.
Now to Cherry. She knows that her two pals don’t have the 4 or the 8. If she has the 4, or the 8, she would state that there are seven possibilities for Apple’s ball, which are all the balls except the 4-ball and the 8-ball. But she states there are four possibilities, so we can rule out the chance she has the 4 or the 8.
So, no one has the 4 or the 8, which means the three pals must have balls from the following selection: 1,2,3,5,6,7 and 9.
From Zog’s statement we can now deduce that these combinations are also impossible: (1,9), (2,6), (5, 9) and (3,7). In each of these cases it is impossible for there to be a third ball such that the number on a ball is equal to the sum of the numbers on the others.
Therefore, if Cherry has a 1, there are five possibilities for Apple (all those except 1, 4, 8 and 9), if Cherry has a 2, there are also five possibilities for Apple (all those except 2, 4,8 and 6), and so on. It is clear that only if Cherry has the 9-ball, will she know that there are 4 possibilities Apple can have, which are 2, 3, 6 and 7, (i.e all those except 1, 4, 5, 8 and 9).
The question asked us to find Cherry’s ball - which is 9!
Apple knows Bean’s ball since its number is is 9 minus the number on her ball, but neither Cherry nor us puzzlers can deduce Bean’s ball.
I hope you enjoyed today’s puzzle, I’ll be back in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
Thanks to Prem Prakash for today’s puzzle. Prem tweets daily puzzles at @1to9Puzzle.
I’m the author of several books of popular maths, including the puzzle books Puzzle Ninja and Can You Solve My Problems? (If you are interested in common knowledge logic puzzles, I write about their history and development in Can You Solve My Problems?)
I also co-write the children’s book series Football School.
