Thread: Moving In the Plane

Suppose you start at the origin. You move 1 unit away from the origin to point A. You want to get back to the origin. However you must take a step 1/2 unit away from point A. Each successive step is 1/2 the distance of your previous step. Can you return to the origin in a finite number of steps?


I was thinking something along the line of showing that it is not possible along a straight line path (line segment from A to the origin) and so it is not possible along any other path. However, I feel like something is off in my reasoning.

Originally Posted by math_science_dude

Suppose you start at the origin. You move 1 unit away from the origin to point A. You want to get back to the origin. However you must take a step 1/2 unit away from point A. Each successive step is 1/2 the distance of your previous step. Can you return to the origin in a finite number of steps?


I was thinking something along the line of showing that it is not possible along a straight line path (line segment from A to the origin) and so it is not possible along any other path. However, I feel like something is off in my reasoning.

You are correct that you cannot return to the origin in a finite number of steps. In fact, if you move by "jumping," you can never return to any of the points you have previously visited.

Originally Posted by math_science_dude

Suppose you start at the origin. You move 1 unit away from the origin to point A. You want to get back to the origin. However you must take a step 1/2 unit away from point A. Each successive step is 1/2 the distance of your previous step. Can you return to the origin in a finite number of steps?


I was thinking something along the line of showing that it is not possible along a straight line path (line segment from A to the origin) and so it is not possible along any other path. However, I feel like something is off in my reasoning.

You can only approach your original point, much like a graph approaches an asymptote. However, if you are only given a finite number of steps, you can only ever get as close as the number of steps you take. In other words, you still be within a finite range of your original location. If you had infinite steps, then theoretically you could reach your original point, but you can't reach it in a finite number of steps.