Problem of the Day #47: Brandon’s Game of LifeMay 5, 2011

Posted by Saketh in : potd , trackback

Brandon has been tinkering with cellular automata. He takes $N$ counters, each of which are black or white, and places them in a ring.

Then, for each pair of adjacent counters in the ring, Brandon places between them a new counter that is black if they have the same color and white otherwise. The original counters are removed to leave a new ring of size $N$.

Brandon wishes to continue performing this operation until all of the counters in the ring have the same color. Find all $N$ for which this will always occur, regardless of the coloring of the original counters.