The Math Forum: 1996 
Summer Institute - sum96

July 15-19, 1996 - Swarthmore, Pennsylvania

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Traffic Jam

As the Math Forum's 1996 Summer Institute began, on Sunday evening, after a few brief introductions to the coming week's activities, Steve Weimar took us out onto the porch of Ashton House, where the floor conveniently consists of a grid of square tiles. Institute participants and staff divided into groups of six each to experiment with the mathematical implications of an activity called Traffic Jam.

Traffic Jam is a game for any even number of people, but it's probably best begun with 1, 2 or 3 on each side. Some of us tried it first with fewer people before going on to experiment with 3 people at each end of the group.
 

Goal:

Equal numbers of people face each other with one open slot between them. Everybody faces the open slot. If there are 6 people there will be 7 slots, 6 of which must always have people in them.

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?

    Here's a link to a Java Applet that runs this puzzle.
    Here's a link to a page of our solutions and reflections on this problem.

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Math Forum * * * * 15 July 1996