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The Way to Weigh


GIVEN: Twelve identically looking coins, except one is imperceptibly lighter or heavier than the remaining eleven.

OBJECTIVE: Using only a balance scale, in three separate weighings determine, which coin it is, and whether it is lighter or heavier.

Note 1: You can play with the problem at Joseph Howard's simulation....be sure to choose 12 coins.

Note 2: Generalize your solution for the problem of having (3n - 3)/2 coins (for n > 1), where one coin is counterfeit and weighs less or more than the other coins?

 

Source: E.A. (Bellingham, WA)


Hint: First, solve an easier version of the problem...where you know one coin is either heavier or lighter than the others.

Or, try it with less than 12 coins...

Then, try to adjust your strategy...

 


Solution Commentary: Did you get it....the final logic is neat but certainly not trivial. Visually, the solution looks like a tree, as each weighing produces 2 or 3 branches (both pans equal, left pan heavier, left pan lighter).

Some web sites that explore the problem's solution and its extensions are a hacker's web site or MAZES.com or high school student's solution.