In some classes, either the instructor or some students bring goodies. One fine day, someone brings a box of the Pot of Gold Cherry Cremes Collection. Each tray looks like:
Suppose that you are handed a tray with three chocolates left in it. Class is over, so you can have them all. Rather than just gobble them all at once, you are going to play with your food. You note that it is possible to jump one chocolate with another and then do it again.
A jump consists of moving a chocolate along a straight line over another chocolate and then onto the next place (which must be empty) and removing the jumped-over chocolate. The starting place, jumped place, and ending place must be three consecutive places in a straight line (e.g. top left, mid top left, and middle).
How many possible arrangements of chocolates are there (at the time you get the tray) where a sequence of two jumps is possible? Consider chocolates to be identical, and that reflections are distinct solutions.
Submit your answer to Gene Wirchenko <email@example.com>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, January 8, 2014 at noon Pacific Time. I will post the answer shortly after.