Given the implementation, an interesting way to cheat is to use the following "never loses" sequence: RRRRRRLRRRLRLRLLRRRRLLRRLRRLRLLLRRRLLLRLRRLLRLRL

Given the implementation, an interesting way to cheat is to use the following "never loses" sequence: RRRRRRLRRRLRLRLLRRRRLLRRLRRLRLLLRRRLLLRLRRLLRLRL RRRRRRLRRRLRLRLLRLLRRRRLLRRLLLLRRLRRLRLLLLLRLLLRRRLLLRLRRLLRLRL RRRRRRLRRRLRLRLLRLLRRRRLLRRLLLLRRLRRLRLLLLLLRLLLRRRLLLRLRRLLRLRL RRRRRRLRRRLRLRLLRLLRRRRLLRRLLLLRRLRRLRLLLLLLRLLLRRRLLLRLRRLLRLRL RRRRRRLRRRLRLRLLRLLRRRRLLRRLLLLRRLRRLRLLLLLLRLLLRRRLLLRLRRLLRLRL ... You could easily avoid the repetition by saying that on ties you just choose a prediction at random (and likewise for the first 4). What's at stake here is the same thing that threatens to eventually cause another civil war in the US, but that is a question for another time. :) It is a broad class of theorems which state that if you have a function f: X -> X and you repeat it over and over it naturally finds an input x such that f(x) = x. This repetition has naturally found a set of moves which resets the Markov matrix back to what it was. If you randomized the prediction on ties then I am not sure what I would do exactly... I would have to steer the Markov matrix into a cycle but assuming that it stores the absolute counts for every single prediction, any oscillations I'm using would want to naturally decay to zero... I might be able to exploit it if you were to only look at, say, my last 1000 moves' history to make your predictions, and maybe I can still induce an oscillation that does better than random chance, but I am not sure that I can.