# 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. *) ToHex[n_] := StringJoin[Map[Function[d, StringTake["0123456789ABCDEF", {d + 1, d + 1}]], IntegerDigits[n, 16]]] ToHex[Sum[15*16^(n-1) - 43*15^(n-1) + 41*14^(n-1) - 13^n, {n, 1, 16}]]