Why a 6 and a 4 can never measure 5

Question

You have a 6-litre jug and a 4-litre jug, both unmarked, plus a tap and a drain. You may fill a jug, empty a jug, or pour one into the other until one is empty or the other is full. A friend insists that with enough patience you can get exactly 5 litres into the 6-litre jug. Is your friend right? If not, give a watertight argument that it is impossible, no matter how many moves you make.

Accepted Answer

It is impossible. The invariant. Both jugs hold an even number of litres at the start (zero each). Now look at every kind of move: - Fill a jug. It now holds 6 or 4 litres: even. - Empty a jug. It holds 0: even. - Pour from one into the other. Either the source empties, moving its whole even content into the target (target still even, source 0), or the target fills to 6 or 4, in which case the source gave away an even amount (the even gap that was in the target) and keeps an even remainder. Either way both jugs remain even. So every reachable state has both jugs holding an even number of litr…

Why a 6 and a 4 can never measure 5

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