Suppose a bug likes to jump around the numbers of a clock, but the bug always skips one number with each jump. Make a graph where the 12 numbers on the clock are the vertices, and two vertices are connected if it is possible for the bug to start at one vertex and end up at the other vertex. If the bug starts on 12 can it reach every other number with its current jumping scheme?
Now suppose the bug always skips 4 numbers with each jump. Make a graph where the 12 numbers on the clock are the vertices, and two vertices are connected if it is possible for the bug to start at one vertex and end up at the other vertex. If the bug starts on 12 can it reach every other number with the new jumping scheme?