You are given 99 coins which consists of 30 fake ones. You also have a digital balance scale with perfect precision that shows how much difference between weighs you put on. For example, if you put 10 g on the left side and 20 g on the other side, it will show -10, otherwise +10.
You are asked to find a fake coin among given 99 coins:
- You know that all genuine coins have the same weight but you do not know their weights.
- You also know that every fake coin is heavier or lighter by 1 gram than any genuine coin.
EDIT: The intended question was to not allow a mix of heavier and lighter coins. Since all answers were based on this assumption, changing this requirement now would invalidate them all. I'll leave this question as is (and allow a mix) but don't know an optimal solution myself.
So, what is the minimum amount of weighing which guarantees to find any fake coin you are looking for? (The fake coin you are going to find might be heavier or lighter, it does not matter, you just need to find any fake one.)
Note: You may assume weights are positive integers, but it is not supposed to change the result.
You may also weigh one or more coins against nothing to get their total weight (originally asked and answered in comments.)
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