![]() The chance that of getting a match on your first turn is $\frac$. In which case, on your second turn you pick a new card, which must be either A or B, allowing you to get a match, so the game take three turns. Alternatively you pick A then B on your first turn. Either you get a match on your first turn, in which case you also get a match on your second turn and the game is over in two turns. With $n = 2$, there are two options for what can happen. You will win in one turn no matter regardless of your strategy. ![]() Starting back at $n = 1$, the game is, again, trivial. Do this for both cards you pick on your turn. Now you have to have a strategy, but the perfect strategy is quite straightforward: if you can make a pair, then make a pair, otherwise turn over a card you haven't previously seen. ![]() Going, from one extreme to the other, what if you have perfect memory (or if you just don't turn the cards back over). ![]()
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