The Ultimate Coin Problem
Most Polymaths are puzzle solvers. It's why they are so curious about so many subjects. So naturally, The Polymath has to have a puzzle column.
We publish puzzles submitted by our members and subscribers. In order to be published, a puzzle must have two characteristics. First, it must be challenging. Second, it must contain at least one "ah ha" point on the way to the solution. By that we mean it must require the solver to see a counter-intuitive solution methodology.
Here's an example:
You have an uncalibrated balance scale and X coins. X-1 of the coins are identical. The remaining coin is identical to the others except that it is either slightly lighter or slightly heavier than the rest. The difference is not noticeable by heft. You must use the scale.
What is the greatest value of X such that the coin of a different weight can be reliably identified in four weighings?
Note: This problem is susceptible to mathematical analysis. However, that will only give you the answer, not the strategy. So if you get the answer by math, imagine X coins in front of you and figure out how to actually do it. Have fun. It's not easy.
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