Dr. Probot is a famous inventor, known for many creations, his most recent being "Bot".
Bot is the very first working robot, he has 3 very basic instructions:
- Self Destruct
Left: Rotate 90° left, move forward one space.
Right: Rotate 90° right, move forward one space.
Self Destruct: Destroy itself for all eternity, leave no traces.
Bots are preprogrammed to pick a random left/right movement. They will also self-destruct if they bump into another bot, or leave the grid.
Each turn all the bots move at once.
The bots are programmed to stop if they are the last one left.
Imagine an $n\times n$ grid full of bots, all facing in one direction. Is there a general strategy to determine if eventually, there will be 1 bot standing, and in how many turns? If there is, what is it? If there isn't, explain why not.