This is a follow-up to the excellent safecracking puzzle based on me misreading the problem. :-)
You have a safe with four switches, each of which can be in one of three states (low, medium, high). Only two of those switches actually do anything; the other two don’t control the safe. As soon as the two correct switches are set to the right values, the safe pops open.
Your goal is to open the safe with as few switch changes as possible. For example, changing the switches from the (low, low, low, low) to (medium, low, low, low) counts as changing one switch’s value. Moving from (low, low, low, low) to (high, low, low, low) counts as two switch changes, since in moving the first switch from low to high we pass through the state of having the first switch set to medium.
What is the fewest number of switch changes that need to be made to guarantee that the safe will open?