September "Bravos" to Steve Weimar, Sarah Seastone, and Sum96 for the suggestions and examples, and Mike Morton for the nifty Applet.

# Traffic Jam

## Activity of the Month

### September 1996

Each month the Math Forum sponsors a student activity that can be done locally and shared globally. There are:

## Traffic Jam

Traffic Jam is a game for any number of people, but it's probably best begun with an even number and 1, 2, or 3 on each side.

### How to Play

Players divide into two groups and line up facing each other with one open slot in the middle:

## 1 2 3 ->     <- 4 5 6

The two groups exchange places, observing the following constraints:

1. If the slot immediately in front of or behind you is empty, you may step into it. There is always only one slot empty.

2. If there's a space on the other side of a person next to you, you may step around that person (but only around one person) into the space.

### Object of the game:

To move from a configuration that starts out as

to:

## 4 5 6   1 2 3

in the most economical way you can find, using the minimum possible number of moves.

### Discussion suggestions:

1. What's the minimum number of moves necessary for two people on a side? for three people on a side?
2. How is this minimum number of moves related to 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. Is there a pattern in the different kinds of moves people make - for instance when they move directly into an empty slot and when they move around another player?
5. Do you find a pattern related to the number of people on a side?
6. Is there a simple set of directions a group can follow to accomplish the task in the minimum number of moves?