"Given a circle of n lights, exactly one of which is initially on, it is permitted to change the state of a bulb provided that one also changes the state of every dth bulb after it (where d is a divisor of n strictly less than n), provided that all n/d bulbs were originally in the same state as one another. Is it possible to turn all the bulbs on by making a sequence of moves of this kind?"
OEIS http://oeis.org/A103314 looks very apropos.
A070894 looks even better. The set of bulbs changed by a move has a center-of-light at zero. (Arrange the bulbs evenly around the circle. And then all go on or all go off.) The original center is non-zero, and a move changes it to a weighted combination of zero and the old value: but never to zero. So you can't even turn them all off. Please refute this, if you're the last one to bed. -- Don Reble djr@nk.ca