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Topic: Summer '96 Institute: Shared Lessons and Activities
Replies: 10   Last Post: Jul 19, 1996 1:30 PM

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Sarah Seastone

Posts: 171
Registered: 12/3/04
Traffic Jam Activity - Math Forum Summer Institute
Posted: Jul 16, 1996 12:38 PM
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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.


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.


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.

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?

At the bottom of the page there's a link to a page of our solutions and
reflections on this
problem with an illustration.

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