Since the switch/lamp-voltage table looks a lot like a truth table, what design hints could a
Karnaugh mapping provide?
The first thing that stands out is that L1 and L5 are identical. This can only happen if they are in series or in parallel. But parallel won't drop enough voltage so they must be in series and connected as a single bulb.
Now look for groups of four. L2 is always off when S1 is open. So L2 must be in series with S1. Both L3 and L4 are always off when S2 is open. So S2 must be in series with L3 and L4.
L4 is always off when S3 is closed. So S3 must be in parallel with L4.
The voltage drops give the series/parallel configuration of the lamps, while the K-map gives the configuration of the switches. It's not a complete design but it should be enough to get to the point where the solution is obvious.