Problem of the Day #112

POTD March 30, 2024

A parking lot consists of 2012 parking spots equally spaced in a line, numbered 1 through 2012. One by one, 2012 cars park in these spots under the following procedure: the first car picks from the 2012 spots uniformly randomly, and each following car picks uniformly randomly among all possible choices which maximize the minimal distance from an already parked car.

If the probability that the last car to park must choose spot 1 can be written as \frac{m}{n}, where m and n are relatively prime positive integers, find m+n.