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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