Don's proof is right (and you'll shortly see the exact same argument in my December blog-essay). So, the one-light-on configurations are NOT accessible from the zero-lights-on configuration. Can anyone devise two configurations that are not mutually accessible even though the turned-on lights have the same center of mass in both configurations? What if we insist that the center of mass in both configurations is the center of the circle? Jim On Friday, December 15, 2017, Don Reble <djr@nk.ca> wrote:
"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
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