Multiple choice exam where no three consecutive answers are the same (1) I wanted to share a cute probability exercise I came up with.

You are given a multiple choice exam. It has twelve questions, and each question has four possible answers labelled (a)-(d).You didn't study for the exam at all, so you might as well just guess the answer to each question. But you do have one important piece of information: the exam was designed so that no three consecutive correct answers have the same label. So if two consecutive correct answers are (c), the next one cannot be (c).
What strategy should adopt to maximize the expected number of questions you answer correctly on the exam, and using this strategy what is the expected number of questions you will answer correctly?

I've posted a continuation of this question here, related to trying to maximize the probability of other "optimal" outcomes.

Assuming that "optimum" just means maximizing the expected number of correct answers, this is one of those problems that is trivialized by linearity of expectation.  For each individual problem, your probability of getting that problem correct is $1/4$, whatever choice you make.  So your expected score on the whole test is $25\%$, no matter how you choose your answers.  The correlations between the answers between the different questions don't matter if you're just looking at the expected value.
A more interesting and probably much harder question would be something like, "how can you maximize the probability that you get at least half the questions correct?".
