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(*
* Solution to Project Euler problem 301
* Copyright (c) Project Nayuki. All rights reserved.
*
* https://www.nayuki.io/page/project-euler-solutions
* https://github.com/nayuki/Project-Euler-solutions
*)
(*
* In a game of Nim where both players play with perfect strategy, if the current state is a collection (multiset) of piles
* with sizes {n1, n2, ..., n_m}, then the current player will lose if and only if n1 XOR n2 XOR ... XOR n_m = 0.
* In this problem, we specialize the condition to just n XOR 2n XOR 3n = 0.
*
* Facts:
* 3n = 2n + n.
* a ^ b = 0 iff a = b. (from digital logic)
* a + b = (a ^ b) + ((a & b) << 1). (from digital logic)
* Hence:
* n ^ 2n ^ 3n = 0 (what we want)
* iff n ^ 2n = 3n (from the second fact)
* iff n ^ 2n = (n ^ 2n) + ((n & 2n) << 1) (from the third fact)
* iff (n & 2n) << 1 = 0 (by cancelling on both sides)
* iff n & 2n = 0 (left-shifting doesn't change zeroness)
* iff the binary representation of n does not have consecutive '1' bits.
*
* How many binary strings of length i have no consecutive 1's?
* First partition the set into strings that begin with a 0 and strings that begin with a 1.
* For those that begin with a 0, the rest of the string can be any string of length i-1 that doesn't have consecutive 1's.
* For those that begin with a 1, the rest of the string can be any string of length i-1 that begins with a 0 and doesn't have consecutive 1's.
* Let numStrings(i, j) be the number of bit strings of length i that begin with the bit j and have no consecutive 1's. Then:
* numStrings(1, 0) = 1. (base case)
* numStrings(1, 1) = 1. (base case)
* numStrings(i, 0) = numStrings(i-1, 0) + numStrings(i-1, 1). (for i >= 2)
* numStrings(i, 1) = numStrings(i-1, 0). (for i >= 2)
* This corresponds to a shifted Fibonacci sequence, because:
* numStrings(i, 0) = numStrings(i-1, 0) + numStrings(i-2, 0). (substitute)
* numStrings(1, 0) = 1. (base case)
* numStrings(2, 0) = 2. (derived)
* So numStrings(i, 0) = fibonacci(i + 1).
* What we want is numStrings(30, 0) + numStrings(30, 1) = numStrings(31, 0) = fibonacci(32).
*
* Actually, that answer considers numbers in the range [0, 2^30), which is not exactly what we want.
* According to the problem statement, we need to exclude 0 and include 2^30. But both are losing positions, so the adjustments cancel out.
*)
Fibonacci[32]