Suppose you have twelve identical-looking balls. Ten of them are the same weight, but one is slightly lighter and one slightly heavier, in such a way that the weight of the lighter and heavier ball add up to exactly the weight of two of the equal-weight balls.
What is the minimum number of weighings on a two-sided scale that are needed to determine which ball is lighter and which one is heavier?
What would the solution be if the weight of the lighter and heavier ball didn't necessarily add up to the weight of two equal balls, but you didn't know whether it was lighter, heavier, or equal to that weight?
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