http://mathforum.org/alejandre/java/jam/Jam.html

People must attempt to exchange places without turning around, so that a configuration that begins as:

-> 1 2 3 4 5 6 <-

will end up:

<- 4 5 6 1 2 3 ->

with everybody facing away from the empty slot in the middle.

Object of the game:

To exchange places in the most economical way that can be found, using the minimum number of possible moves.

Rules:

1. If a space in front of you is empty, you may move forward into it. (If a space behind you is empty, you may move backward, but this will probably not be the most economical way to accomplish your goal.) We called this move a slide.

2. If there's an empty space in front of the person directly in front of you, you may jump the person in front of you and move into the empty space unless you're both facing in the same direction (players facing in the same direction may not jump each other).

3. You may not turn around.

Questions considered by the whole group:

1. What's the minimum number of moves necessary for two people on a side? for three people on a side?
2. What's the pattern in the number of moves it takes and the number of people on a side?
3. What formula can account for the minimum number of moves for any number of people on a side?
4. What pattern of 'slides' and 'jumps' can be found, and how is it related to the number of people on a side?
5. What directions does a group follow to accomplish the task with the minimum number of moves?