### July 15-19, 1996 - Swarthmore, Pennsylvania

Back to Traffic Jam || 1996 Summer Institute Main Page || Agenda

## Traffic Jam Solutions and Reflections

### One group's investigations

Elizabeth Weber describes her group:
My group tried it a few times and we found we could get from )))_((( to (((_))) easily, but it involved 20 steps - someone had to step backward and then forward again. We wondered if we could improve on this and tried the problem with two people, going from )_( to (_). No problem - 3 steps, no backing up. Then we tried it with 4 people. This took a little more work, but we figured out how to go from ))_(( to ((_)) in 8 steps wihout anyone ever stepping backward. Then we went back to the six-person version.

It took about 20 minutes, more or less, but we got rid of the backward step and were able to solve the problem in 15 moves. Since no one ever moved backward or changed places with anyone facing the same direction, we figured this was the minimum number of steps.

Our method was mostly trial and error. The only rule we figured out was to try to weave the people facing in opposite directions: after you started moving, you should try to keep yourself between people facing in the opposite direction. We were going to figure out exactly what the pattern was - maybe it had something to do with the position of the empty slot? - but just then Steve called us all back inside to discuss what we'd done.

### Reflections

After a half-hour experimenting, everybody went back inside to discuss the math. Here are some ideas that surfaced:
• Never slide so that two people heading in the same direction will be next to each other until they arrive in their final position.

• If you start with three people on a side, everyone moves a total distance of four spaces. The group has to travel a minimum of 24 spaces. In our fifteen moves we did nine jumps, each of which covered two spaces (18 total), and six slides. 18 + 6 = 24 so we convinced ourselves that our pattern of movement of 9 jumps + 6 slides was as good as we could do.

• If n = the number of people on a side, then n^2 + 2n = the number of moves.

• The number of jumps equals n^2 and the number of slides equals 2n and looking at slides and jumps is one way to come to a formula. Participants who think aurally and physically bring an important perspective.

• There's an interesting distribution of slides and jumps.
one person on a side: sjs
two persons on a side: sjsjjsjs
three persons on a side: sjsjjsjjjsjjsjs

Traffic Jam is an interesting exercise that highlights aspects of effective cooperation and can be used to focus on the development of different representations of the problem, and different languages (visual, symbolic, etc.) that bring out varied ways of perceiving and working through a problem.

### A graphical representation

Here's one way to show the moves for three people on a side:

 Everybody faces the center space: Green slides forward into the open space. Yellow jumps green. Orange slides forward. Green jumps orange. Blue jumps yellow. Purple slides forward. Yellow jumps purple to reach the end of the row. Orange jumps blue. Red jumps green. Green slides forward to reach the end of the row. Blue jumps red. Purple jumps orange. Orange slides forward. Red jumps purple. Purple slides forward.

- Sarah Seastone