Suppose at the start of the game, we flip the cards in positions i and j and find that they match! These cards thus remain face-up for the rest of the game (and we observe their movement under the mystery permutation \sigma). Let’s see how much information we can get out of this.
For the problems in this section, suppose that the indices i and j are part of cycles C_{i} and C_{j} in \sigma, which have lengths c_{i} and c_{j}, respectively, that we are trying to figure out. (To determine C_{i} means to figure out the indices in C_{i}; equivalently, to compute \sigma^{n}(i) for any integer n.)
Find a strategy which guarantees at least one of the following two things:
- the strategy proves that c_{i}=c_{j}, or
- the strategy proves that c_{i} \neq c_{j}, determines the value of \min \left\{c_{i}, c_{j}\right\}, and determines which of c_{i} and c_{j} is smaller.