Splay Trees

Level pending

The number \(\frac{1}{2}\) is written on a blackboard. For a real number \(c\) with \(0 < c < 1\), a \(c\)-splay is an operation in which every number \(x\) on the board is erased and replaced by the two numbers \(cx\) and \(1-c(1-x)\). A splay-sequence \(C = (c_1,c_2,c_3,c_4)\)is an application of a \(c_i\)-splay for \(i=1,2,3,4\) in that order, and its power is defined by \(P(C) = c_1c_2c_3c_4\).

Let \(S\) be the set of splay-sequences which yield the numbers \(\frac{1}{257}, \frac{2}{257}, \dots, \frac{256}{256}\) on the blackboard in some order. If \(\sum_{C \in S} P(C) = \tfrac mn\) for relatively prime positive integers \(m\)and \(n\), compute \(100m+n\).

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