There are seven lights in a circle. They are all turned off and the problem is to turn them on.

Each light has a switch. The trouble is that the electrician didn't use proper switches. When you press a switch, if the light is on it turns off, and if a light is off it turns on as usual, however, it also does the same for the two lights on either side.

So, if you pressed the switch for light B in the left picture, lights A and B would turn on and light C would turn off as shown in the right picture.

Extension:

Is it possible to turn the lights all off if you start with 6 lights, one of which is on?

Let's group together all of the on-off configurations that can be obtained from switching on and off lights. How many different groups are there for the seven-in-a-circle and six-in-a-circle arrangements?

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