# Problem 503: Compromise or persist
Alice is playing a game with n cards numbered 1 to n. A game consists of
iterations of the following steps. (1) Alice picks one of the cards at
random. (2) Alice cannot see the number on it. Instead, Bob, one of her
friends, sees the number and tells Alice how many previously-seen
numbers are bigger than the number which he is seeing. (3) Alice can end
or continue the game. If she decides to end, the number becomes her
score. If she decides to continue, the card is removed from the game and
she returns to (1). If there is no card left, she is forced to end the
game. Let F(n) be the Alice's expected score if she takes the optimized
strategy to minimize her score. For example, F(3) = 5/3. At the first
iteration, she should continue the game. At the second iteration, she
should end the game if Bob says that one previously-seen number is
bigger than the number which he is seeing, otherwise she should continue
the game. We can also verify that F(4) = 15/8 and F(10) ≈ 2.5579365079.
Find F(106). Give your answer rounded to 10 decimal places behind the
decimal point.