# nayuki/Project-Euler-solutions

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 # # Solution to Project Euler problem 162 # Copyright (c) Project Nayuki. All rights reserved. # # https://www.nayuki.io/page/project-euler-solutions # https://github.com/nayuki/Project-Euler-solutions # # Among the set of n-digit hexadecimal numbers, how many of them: # # Are there in total?: 15*16^(n-1). # # Have no 0?: 15^n. # Have no 1?: 14*15^(n-1). # Have no A?: 14*15^(n-1). # # Have some 0?: 15*16^(n-1) - 15^n. # Have some 1?: 15*16^(n-1) - 14*15^(n-1). # Have some A?: 15*16^(n-1) - 14*15^(n-1). # # Have no 0 and have no 1?: 14^n. # Have no 0 and have no A?: 14^n. # Have no 1 and have no A?: 13*14^(n-1). # # Have some 0 or have some 1: 15*16^(n-1) - 14^n. # Have some 0 or have some A: 15*16^(n-1) - 14^n. # Have some 1 or have some A: 15*16^(n-1) - 13*14^(n-1). # # Note: (Have X) + (Have Y) = (Have X or have Y) + (Have X and have Y). # Have some 0 and have some 1: (15*16^(n-1) - 15^n) + (15*16^(n-1) - 14*15^(n-1)) - (15*16^(n-1) - 14^n) = 15*16^(n-1) - 29*15^(n-1) + 14^n. # Have some 0 and have some A: (15*16^(n-1) - 15^n) + (15*16^(n-1) - 14*15^(n-1)) - (15*16^(n-1) - 14^n) = 15*16^(n-1) - 29*15^(n-1) + 14^n. # Have some 1 and have some A: (15*16^(n-1) - 14*15^(n-1)) + (15*16^(n-1) - 14*15^(n-1)) - (15*16^(n-1) - 13*14^(n-1)) = 15*16^(n-1) - 28*15^(n-1) + 13*14^(n-1). # # Have no 0 and have no 1 and have no A? : 13^n. # Have some 0 or have some 1 or have some A?: 15*16^(n-1) - 13^n. # # Note: (Have 0 or have 1 or have A) = (Have 0) + (Have 1) + (Have A) - (Have 0 and have 1) - (Have 0 and have A) - (Have 1 and have A) + (Have 0 and have 1 and have A). # Therefore: # Have 0 and have 1 and have A # = (15*16^(n-1) - 13^n) - (15*16^(n-1) - 15^n) - (15*16^(n-1) - 14*15^(n-1)) - (15*16^(n-1) - 14*15^(n-1)) + (15*16^(n-1) - 29*15^(n-1) + 14^n) + (15*16^(n-1) - 29*15^(n-1) + 14^n) + (15*16^(n-1) - 28*15^(n-1) + 13*14^(n-1)) # = 15*16^(n-1) - 43*15^(n-1) + 41*14^(n-1) - 13^n. def compute(): ans = sum((15 * 16**(n-1) - 43 * 15**(n-1) + 41 * 14**(n-1) - 13**n) for n in range(1, 17)) return "{:X}".format(ans) if __name__ == "__main__": print(compute())