Answer to Problem of the Week for 09/11/2000

Answer to September 11, 2000 Problem

by Dr. Michael Ecker, Recreational & Educational Computing editor

The Spiraling Bug Problem

A "bug" of negligible dimensions starts at the origin(0,0) of the standard two-dimensional rectangular coordinate system. The bug walks one unit right, then one-half unit up, then 1/4 unit left, then 1/8 unit down, etc. In each move, it always turns counter-clockwise at a 90 degree angle and goes half the distance it went on the previous move. Which point (x,y) in the xy-plane is the bug approaching in its spiraling journey?
Solution to Problem: Answer is (4/5, 2/5) Solution: You can think of the x and y coordinates as each being the sum of an infinite geometric series. x = 1 - 1/4 + 1/16 - 1/64 + 1/256 + ... Using the formula for the sum of an infinite geometric series, S = a1 / (1 - r) where a1 = 1 (the first term) and r = -1/4 (common ratio), S = 1 / (1 - (-1/4)) = 1 / (5/4) = 4/5 Similarly, y = 1/2 - 1/8 + 1/32 - 1/128 + ... So, S = (1/2) / (1 + 1/4) = 2/5

A "bug" of negligible dimensions starts at the origin(0,0) of the standard two-dimensional rectangular coordinate system. The bug walks one unit right, then one-half unit up, then 1/4 unit left, then 1/8 unit down, etc.
In each move, it always turns counter-clockwise at a 90 degree angle and goes half the distance it went on the previous move.

Which point (x,y) in the xy-plane is the bug approaching in its spiraling journey?

Solution to Problem:

Answer is (4/5, 2/5)

Solution:
You can think of the x and y coordinates as each being the sum of an infinite geometric series.
x = 1 - 1/4 + 1/16 - 1/64 + 1/256 + ...
Using the formula for the sum of an infinite geometric series,
S = a1 / (1 - r) where a1 = 1 (the first term) and r = -1/4 (common ratio),
S = 1 / (1 - (-1/4)) = 1 / (5/4) = 4/5
Similarly, y = 1/2 - 1/8 + 1/32 - 1/128 + ...
So, S = (1/2) / (1 + 1/4) = 2/5

Correctly solved by:

1. Bob Hearn	Winchester, VA
2. Chip Schweikarth	Winchester, VA
3. Jonathan Pence	Virginia Tech