Puzzle 1: There are three water jugs, which can hold 8, 5 and 3 gallons of water respectively. The first jug is full, and the other two are empty. It is necessary to divide the water into two equal portions, that is, to put exactly four gallons into both the 8 and 5 gallon jugs. The only operation allowed is to pour some or all of the contents of one jug into another.
Can the water be divided into two equal portions? How?
Puzzle 2: Under the same conditions as Puzzle 1, we now want to divide the water into portions of 4, 2 and 2 gallons? Can this be done? How?
Puzzle 3: Robinson Crusoe has asked you to visit him on his desert island. He owns a 12 gallon jug, and is very depressed that he can only prepare recipes that call for exactly 12 gallons of water. You decide to bring another measuring jug with you. At the store, the choices are every integral value from 1 to 12 gallons. What size jug should you buy so that Robinson Crusoe will be able to measure out the most number of different amounts of water?
That is, assume an unlimited supply of water, and allow Crusoe to transfer the contents of either jug to the other (up to the filling point), and to "top up" either jug at any time with more water. To "measure out" N gallons of water is to arrive at a point at which you can state that the sum of the contents of the two jugs is N.
At the moment, Crusoe can only measure out 0 and 12 gallons. If you buy Crusoe a 1 gallon jug, he will be able to measure out 0, 1, 2, 3, ..., 12 and 13 gallons, for a total of 14 different values. Can you do better?
I give up, show me the solution.