Merge pull request #34 from georgenetu/master

Added Problem 481

README.md

@@ -12347,3 +12347,23 @@ Give your answer using lowercase characters (no punctuation or space).
 
 Answer: b97e157bf53033d21f610d2350b92faf
 
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+Problem 481
+===========
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+
+A group of chefs (numbered #1, #2, etc) participate in a turn-based strategic cooking competition. On each chef's turn, he/she cooks up a dish to the best of his/her ability and gives it to a separate panel of judges for taste-testing. Let S(_k_) represent chef #_k_'s skill level (which is publicly known). More specifically, S(_k_) is the probability that chef #_k_'s dish will be assessed favorably by the judges (on any/all turns). If the dish receives a favorable rating, then the chef must choose one other chef to be eliminated from the competition. The last chef remaining in the competition is the winner.
+
+The game always begins with chef #1, with the turn order iterating sequentially over the rest of the chefs still in play. Then the cycle repeats from the lowest-numbered chef. All chefs aim to optimize their chances of winning within the rules as stated, assuming that the other chefs behave in the same manner. In the event that a chef has more than one equally-optimal elimination choice, assume that the chosen chef is always the one with the next-closest turn.
+
+Define W<sub>n</sub>(_k_) as the probability that chef #_k_ wins in a competition with n chefs. If we have S(1) = 0.25, S(2) = 0.5, and S(3) = 1, then W3(1) = 0.29375.
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+Going forward, we assign S(_k_) = F<sub>k</sub>/F<sub>n+1</sub> over all 1 ≤ _k_ ≤ _n_, where F<sub>k</sub> is a Fibonacci number: F<sub>k</sub> = F<sub>k-1</sub> + F<sub>k-2</sub> with base cases F<sub>1</sub> = F<sub>2</sub> = 1. Then, for example, when considering a competition with _n_ = 7 chefs, we have W<sub>7</sub>(1) = 0.08965042, W<sub>7</sub>(2) = 0.20775702, W<sub>7</sub>(3) = 0.15291406, W<sub>7</sub>(4) = 0.14554098, W<sub>7</sub>(5) = 0.15905291, W<sub>7</sub>(6) = 0.10261412, and W<sub>7</sub>(7) = 0.14247050, rounded to 8 decimal places each.
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+Let E(_n_) represent the expected number of dishes cooked in a competition with _n_ chefs. For instance, E(7) = 42.28176050.
+
+Find E(14) rounded to 8 decimal places.
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+Answer: 2dfd5108fcb120426ae938aed3bfef8f