The Empty Bottle

  1. Consider three bottles of water. The first contains 1 ml, the second 5 ml and the third 10 ml. Each bottle can contain the total volume (16 ml in this case). The rule is that we can pour the content of a bottle into another one only if we can double the content of the second one. For example, we can pour 5 ml from the third bottle into the second one, or 1 ml from either of the last two bottles into the first one. Show that it is possible to get very easily and quickly an empty bottle. What is the minimum number of pourings necessary?
  2. Repeat for volumes of 3, 8, and 23 ml.
  3. Repeat for large volumes of 37, 91, and 203 ml.
  4. Repeat for positive integer volumes of a, b, and c ml, c>b>a. Can an empty bottle always be reached? What requirements are on a, b, and c?

Source: Reader Philippe Fondanaiche.

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