Guessing Game (noch nicht übersetzt)

Problem 406

We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions. Each number (question) we ask, we get one of three possible answers:

  • "Your guess is lower than the hidden number" (and you incur a cost of a), or
  • "Your guess is higher than the hidden number" (and you incur a cost of b), or
  • "Yes, that's it!" (and the game ends).

Given the value of n, a, and b, an optimal strategy minimizes the total cost for the worst possible case.

For example, if n = 5, a = 2, and b = 3, then we may begin by asking "2" as our first question.

If we are told that 2 is higher than the hidden number (for a cost of b=3), then we are sure that "1" is the hidden number (for a total cost of 3).
If we are told that 2 is lower than the hidden number (for a cost of a=2), then our next question will be "4".
If we are told that 4 is higher than the hidden number (for a cost of b=3), then we are sure that "3" is the hidden number (for a total cost of 2+3=5).
If we are told that 4 is lower than the hidden number (for a cost of a=2), then we are sure that "5" is the hidden number (for a total cost of 2+2=4).
Thus, the worst-case cost achieved by this strategy is 5. It can also be shown that this is the lowest worst-case cost that can be achieved. So, in fact, we have just described an optimal strategy for the given values of n, a, and b.

Let C(n,a,b) be the worst-case cost achieved by an optimal strategy for the given values of n, a and b.

Here are a few examples:
C(5,2,3)=5
C(500,2,3)=13.22073197
C(20000,5,7)=82
C(2000000,5,7)=49.63755955

Let Fk be the Fibonacci numbers: Fk=Fk1+Fk2 with base cases F1=F2=1.
Find 30k=1C(1012,k,Fk), and give your answer rounded to 8 decimal places behind the decimal point.