From New Scientist #2293, 2nd June 2001 [link]
There are 8 lights, A, B, C, D, E, F, G, H, in the junior dormitory and each one has its own switch. To save matron having to operate all 8 switches, 12 pairs of lights are connected. They are AB, AD, BC, BE, CF, CH, DF, DG, EF, EG, EH, GH.
Whenever a switch is operated it changes its own light, and each of the lights connected to it, from on to off or from off to on. When matron enters the dormitory at 9.00 pm, lights B, C, E and G are on and the other four lights are off. As an example, if she operates switch F then lights D and F come on and lights C and E go off. If she then operates switch E the lights E and H come on and lights B, F and G go off.
Question 1. Is it possible for matron to operate certain switches so that, when she has finished, all the lights are off? If it is possible, which switches should she operate?
If it is not possible, which one of the four lights that were off at 9.00 pm should additionally have been on at 9.00 pm so that matron could have operated certain switches and finished with all the lights off?
The situation is similar in the senior dormitory except that there are 100 lights and 400 pairs of lights are connected and all the lights are on at 9.00 pm.
Question 2. Can we say for certain that matron can find a selection of switches to operate so that when she has finished all the lights are off?