How many balls must you select to be assured of selecting a black ball?

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How many must you select to be assured of getting a white ball?

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How many must you select to be assured of getting all the black balls?

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Now that you have all 81 black balls, you are told that they are not truly identical, but that one is slightly heavier than the others. You are given a set of balance scales and instructed to find the heavy ball. What is the minimum number of weighings in which you can guarantee to find the heavy ball?

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Using the above method, what is the *minimum* number of weighings in which you could actually find the heavy ball?

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Eighteen of the nineteen white balls are identical. The nineteenth is of a slightly different weight, but you don't know if it's heavier or lighter (and can't tell by feel). Using the balance scales, what is your weighing strategy for finding the odd ball and what is the minimum number of weighings to guarantee finding it? Assume that the eighteen balls are lettered from A to R so that you can tell them apart. (I don't have the optimal solution to this, since I just made it up and it's not obvious to me, but I'll think about it while reading other's responses.)