Can you solve it? The mystery of Cherry's lottery ball

A logic puzzle about knowledge

Update: Several readers spotted a mistake in the setting of the puzzle. Apologies. (Prem, who set the puzzle, responds below the line). In order to turn the puzzle into one that contains no errors, the conversation should go as follows: Apple: “I see 8 possibilities for Bean’s ball”, Bean: “I see 7 possibilities for Cherry’s ball”, Cherry: “I see 5 possibilities for Apple’s ball”, Apple: “I see 3 possibilities for Bean’s ball”, B: “I know Cherry’s ball”, Cherry: “I know Apple’s ball”. What is Apple’s ball?

Hi guzzlers

Of the many types of puzzle I have posed over the years, the genre that readers seem to respond to best are “common knowledge” logic puzzles. In these problems, there is a situation involving at least two people, each of whom has incomplete information about each other, and the solver makes successive deductions hopping back and forth from each person’s point of view. There is usually a charm to the set-up and even though no technical mathematics is used they are very brain-twisty!

Here’s a common knowledge puzzle involving lottery balls.

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.

I’ll be back at 5pm with the solution, and an explanation of the solution.

Meanwhile, NO SPOILERS.

UPDATE: solution is available here.

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.

Puzzle Ninja
Puzzle Ninja Photograph: Guardian Faber

I’m the author of several books of popular maths, including the puzzle books Puzzle Ninja and Can You Solve My Problems?

I also co-write the children’s book series Football School.

Contributor

Alex Bellos

The GuardianTramp

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