Three Containers Puzzle

You have three containers. The 3 L container (L means liter) is empty and the 5 L and 6 L containers are full with 50° and 90° water, as shown initially in this Demonstration. We will refer to them as containers 1, 2, and 3, respectively. By transferring water from one container to another we change the amount and temperature of their contents. For instance, if we transfer from container 3 to 1 and then from 2 to 3, we end up with container 1 full with 90° water, container 2 with 2 L of 50° water, and container 3 full with 70° water. The temperature of container 3 changed from 90° to 70° because 3 L of 90° water and 3 L of 50° water produce 6 L of water at 70° (70 being the average of 90 and 50). If we now fill container 2, it will hold water at 62° because mixing 3 L of 70° water and 2 L of 50° water produces 5 L of 62° water, 62 being equal to (3×70 + 2×50)/(3 + 2).
So, the problem of obtaining some 62° water is solved in three steps: 31, 23, and 32. Similarly, to obtain some (it does not matter how much) 70° water, we need two steps: 31 and 23.
Can you obtain some 74° water? (You need only two steps.) The other possible water temperatures you can obtain are, in order of difficulty, 82°, 66°, 80°, 71°, 72°, and 78°, the last one being the hardest, involving six steps. Try to solve the puzzle for any of these temperatures using at most six steps. If you are stuck, click the "RESET" button to take you to the initial state of the puzzle. The snapshots show the steps for 71° water.


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