The following puzzle was told to me by a friend, Markus Götz, who put it online here: Deviating Ball Puzzles (pdf). After some searching, I did not find this puzzle anywhere else online. The only similar (but ultimately different) puzzle I found is Three Pan Balance, 15 Coins.
Are there any other of three-pan balance puzzles online?
The three-pan balance
Imagine a balance with not two, but three pans. Weightings using the balance follow these rules:
- If there exists a pan that is lighter than each of the other two pans, then this pan goes up and the other two pans go down to a stop. (Note that one cannot see which of the two heavier pans, if any, is the heaviest.)
- If there is no single lightest pan, then nothing happens. (This includes the case of two equally light pans and one heavier pan.)
Let's call this the "lightest-pan-detection-rule" (LPDR).
The puzzle consists of several related problems. (The original contains 12 problems, with different kinds of ball deviation. I will split them across several questions.)
In each problem you are given n balls. They are all of the same weight, exept as stated in each problem. You are to identify the deviating ball(s) by using the new balance a maximum number of weighings stated in the puzzle. You are also to present a method to identify the deviating ball(s).
1) You are given n balls, one of which is lighter. What is the largest n, so that the lighter ball can be identified with 1 weighting?
2) You are given n balls, one of which is lighter. What is the largest n, so that the lighter ball can be identified with 2 weightings?
3) You are given n balls, one of which is lighter. What is the largest n, so that the lighter ball can be identified with k weightings?