The stepmother, leaving for the ball, gave Cinderella a bag in which there were mixed poppy and millet, and ordered her to sort them. When Cinderella left for the ball, she left three bags: one with millet, another with poppy, and a third with the unsorted mixture. To avoid mixing up the bags, Cinderella attached a label to each: "Poppy," "Millet," and "Mixture." The stepmother returned from the ball first and deliberately switched all the labels so that each bag had an incorrect label. The Fairy's student managed to warn Cinderella that now none of the labels on the bags corresponded to the contents. Cinderella then took only one single grain from one bag and, by looking at it, immediately figured out what was in each bag. How did she do it?
Think about whether you can take a grain from the bag labeled "Poppy".

## Solution

You should take a grain from the bag labeled "Mixture." It cannot contain a mixture, so it must contain the grains that we take from it. Let's assume for definiteness that this bag contains poppy. (This assumption is also made for convenience; as an exercise, try repeating all the reasoning for the case where the bag labeled "Mixture" contains millet.) So, the bag labeled "Mixture" contains poppy. This means that the bag labeled "Poppy" can only contain millet (if it contained a mixture, then the bag labeled "Millet" would contain millet, which is impossible). From this, it immediately follows that the bag labeled "Millet" contains the mixture.

## Answer

You should take a grain from the bag labeled "Mixture."

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