A. Garland

Note that there are only a few configuration classes: 1111, 1112, 1122, 1123 and 1234. Let's discuss each of them.

If all 4 bulbs are of the same color, then it is impossible to turn all the bulbs on, because after you switch one light bulb, it is impossible to turn the others on.

If there is a color with 3 bulbs, then it is impossible to turn all the bulbs on in 4 operations, which means there is a bulb that turns on, turns off and then turns on again, i.e. the answer is at least 6 operations. And there is a sequence of exactly 6 operations (such an example was shown in the problem notes).

For configurations like 1122 and 1123, it is enough to turn on the 1 color bulbs not in a row (i.e. in order [1,2,1,2] for the first case and [1,2,1,3] for the second one). So the answer for such configurations is 4 .

If all the bulbs are of different colors, then nothing prevents you from turning them all on in 4 operations.