Juku has four cans of juice: a -litre can containing litres of juice, a -litre can containing litres of juice, a -litre can containing litres of juice and a -litre can containing litres of juice. There are no volume markings on the cans. Juku wants to measure exactly litres of juice by a sequence of pourings of the juice from one can to another. Letting the juice spill is not allowed. Find out all cans into which Juku can get the desired exact amount of juice.
Solution
Answer: The second, the third, the fourth.
The second can contains litres of free space. Pouring juice from the fourth can over to the second can until the second can becomes full leaves litres of juice in the fourth can. If, after that, one pours all juice from either the second or the third can over to the first can (this is possible since the first can contains litres of free space and none of the other cans can contain more juice), then the juice in the fourth can can be poured over to either the second or the third can. Thus Juku can get exactly litres of juice into the fourth can, as well as into the third or the second one.
On the other hand, the total amount of juice in all cans is litres after any sequence of pourings. Thus even if the second, the third and the fourth can are full, the first can contains litres of juice; if the other cans contain free space then the amount of juice in the first can must be even larger. Consequently, it is impossible to get exactly litres of juice into the first can.
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