# Catch that mouse!

A mouse lives inside a wall and has three holes that he travels between. You have a flashlight and want to shine it on the mouse.

At time $$t = 1$$ the mouse will be at one of the holes with probability $$\frac{1}{3}$$. For each integer $$i > 1,$$ at time $$t = i,$$ the mouse moves to a hole different from the one it was at for time $$t = i-1,$$ with probability $$\frac{1}{2}$$ of moving to either hole.

For each integer time $$t = 1, 2, \ldots,$$ you choose one of the three holes and shine your light in it. Let $$E$$ be the expected time you take to shine your flashlight onto the mouse for the first time. Using an optimal strategy, the smallest possible value of $$E$$ can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b?$$

Details and assumptions

If $$E$$ is an integer, then $$\frac{a}{b} = \frac{E}{1}.$$

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