yaworsw/euler-manager

---
:id: 406
:name: Guessing Game
:url: https://projecteuler.net/problem=406
:content: "We are trying to find a hidden number selected from the set of integers
  {1, 2, ..., <var>n</var>} by asking questions. Each number (question) we ask, we
  get one of three possible answers:\n\n- \"Your guess is lower than the hidden number\"
  (and you incur a cost of <var>a</var>), or\n- \"Your guess is higher than the hidden
  number\" (and you incur a cost of <var>b</var>), or\n- \"Yes, that's it!\" (and
  the game ends).\n\nGiven the value of <var>n</var>, <var>a</var>, and <var>b</var>,
  an _optimal strategy_ minimizes the total cost <u>for the worst possible case</u>.\n\nFor
  example, if <var>n</var> = 5, <var>a</var> = 2, and <var>b</var> = 3, then we may
  begin by asking \" **2**\" as our first question.\n\nIf we are told that 2 is higher
  than the hidden number (for a cost of <var>b</var>=3), then we are sure that \"
  **1**\" is the hidden number (for a total cost of **3** ).  \nIf we are told that
  2 is lower than the hidden number (for a cost of <var>a</var>=2), then our next
  question will be \" **4**\".  \nIf we are told that 4 is higher than the hidden
  number (for a cost of <var>b</var>=3), then we are sure that \" **3**\" is the hidden
  number (for a total cost of 2+3= **5** ).  \nIf we are told that 4 is lower than
  the hidden number (for a cost of <var>a</var>=2), then we are sure that \" **5**\"
  is the hidden number (for a total cost of 2+2= **4** ).  \nThus, 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 <var>n</var>, <var>a</var>, and <var>b</var>.\n\nLet
  C(<var>n</var>, <var>a</var>, <var>b</var>) be the worst-case cost achieved by an
  optimal strategy for the given values of <var>n</var>, <var>a</var>, and <var>b</var>.\n\nHere
  are a few examples:  \nC(5, 2, 3) = 5  \nC(500, √2, √3) = 13.22073197...  \nC(20000,
  5, 7) = 82  \nC(2000000, √5, √7) = 49.63755955...\n\nLet F<sub><var>k</var></sub>
  be the Fibonacci numbers: F<sub><var>k</var></sub> = F<sub><var>k</var>-1</sub>
  + F<sub><var>k</var>-2</sub> with base cases F<sub>1</sub> = F<sub>2</sub> = 1.
  \ \nFind ∑<sub>1≤<var>k</var>≤30</sub>&nbsp;C(10<sup>12</sup>, √<var>k</var>, √F<sub><var>k</var></sub>),
  and give your answer rounded to 8 decimal places behind the decimal point.\n\n"