Problem:
Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at and miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are miles from Dodge, and they have been traveling for minutes. Find .
Solution:
Sparky, Butch, and Sundance take and minutes, respectively, to walk one mile. Let be the number of miles Butch walks. Then he rides miles and needs a total of minutes to cover miles. Similarly, Sundance needs minutes. Equating these expressions gives . Because and are positive integers, the smallest solutions are and . Thus Butch walks miles and rides miles, taking minutes to cover miles, so .
The problems on this page are the property of the MAA's American Mathematics Competitions