Walking a Line

21. Galactic Gridlock

Walking a Line

Imagine yourself balanced on a tightrope high above the ground. You can take steps only forward or backward.

Similarly, in a one-dimensional random walk, the walker is confined to a long, narrow path and can step in either of two directions. Tossing a coin would be one way to determine randomly whether to step forward (heads) or backward (tails).

You can keep track of a one-dimensional random walk by plotting a graph that shows how far away you are from your starting point after each toss of a coin.

If your first toss is heads, you end up one step forward from the start. If you toss heads again, you end up two steps forward. A third toss of heads would bring you three steps forward. If your fourth toss were tails, you would step back to the spot two steps away from the start.

This graph shows the results of a one-dimensional random walk. The horizontal axis (x axis)  represents the number of steps taken, and the vertical axis (y axis) shows how many steps you are away from your starting point if you start at zero. Steps in the forward direction are positive (upward) and steps backward are negative (downward).

TRY IT!

Create your own random walk in one dimension.

You will need:

• pencil and paper
• ruler
• coin
• button or some other object to represent the walker

What to do:

1. Using your ruler, draw a line about 12 inches (30 centimeters) long.
2. Mark a starting point in the middle of your line.
3. Choose the length of your walker's "step." One inch or one centimeter would be a good length. Using your ruler, mark off several steps to the right of your starting point and several steps to the left. Start your walker at the midpoint.
4. Flip the coin. If it lands on heads, move your walker to the right from your starting point. If it lands on tails, move it one step to the left.
5. Flip the coin again and go one step to the right for heads or one step to the left for tails, starting from the point where your walker landed after the previous coin flip.
6. Continue flipping the coin and marking the walker position.
7. What do you observe about the walker's position after a number of coin tosses? Try the exercise again, starting from the original starting point. Do you get the same sort of result?

The more times you flip the coin, the farther the walker is likely to stray from the original starting point. Mathematicians have shown that the most probable distance (in steps) from the start equals the square root of the number of steps taken to get to that point.

In other words, after tossing the coin nine times, the walker is likely to be the square root of nine, or three, steps from the start.

Of course, your walker may be two steps, four steps, or some other distance from the start, but if you repeat the experiment enough times, the walker's most probable distance from the start (after each nine tosses of the coin) will be three steps.

Does you walker at any time return to its original starting point? Mathematicians have proved that a random walker going back and forth along a line will eventually return to the start.

That might sound like a good strategy for anyone who is lost on a tightrope: just take steps at random in either direction, and you will end up where you started, though it could take longer than a lifetime to get there!

NEXT: Walking on a Grid