This word problem came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math, and we weren't sure how to get the answer. The word problem goes like this:
Grandma Q drove her car downtown to do some shopping and parked it at a random spot on the street. While shopping she forgot where she parked and called me on her cell phone asking how to find her car.
The downtown is a 10x10 set of square city blocks in a regular grid pattern. The car could be on any one of the streets of this grid, including the outside edges, between two intersections.
Grandma Q can see both sides of the street so she only has to walk one time between any two intersections to search for her car. However she can only see cars on the street she is on between the two intersections she is walking (i.e. she can't look down to the next block or from an intersection to the four blocks it connects).
Because she is elderly and gets tired easily (but not while shopping apparently), I don't want to have her walk where she has already searched, and I want to get to the car as quickly as possible.
Is there a mathematical way to determine the most efficient (minimum time, minimum retracing) search pattern for Grandma Q? Is there more than one solution? Does it matter where she starts (center, edge, corner, random)? Can the answer be scaled to any x-by-x square or x-by-y rectangle?
(we weren't sure which branch of math most effectively deals with this kind of problem, so if someone could tell me, that would be great too.)