There is a clock face lying on my table, upon which I have placed cheese on all of the numbers, one through twelve. I have trained a mouse to start by eating all of the cheese from whichever number it pleases, and then to eat all of the cheese of any other number, so long as the new number still has cheese and is either a multiple of, or a divisor of, the number from which it has just come. The mouse continues to run to new numbers with the same restriction. It can never go back to the same number (no cheese there).
My mouse is now smart enough to eat the maximum number of pieces of cheese that it can. How many pieces is that, and what is the order it takes (there are a few solutions for the maximum, not counting just flipping the entire route).