# Permutations: Alternative Solutions ## Odd \( n \), Start With \( n \) Example: \( 5, 3, 1, 4, 2 \). General: \( n, \dots, 1, (n - 2), \dots, 2 \). Works for: \( (n - 2) \ge 3 \implies n \ge 5 \). ## Odd \( n \), Start With \( n - 1 \) Example: \( 4, 2, 5, 3, 1 \). General: \( (n - 1), \dots, 2, n, \dots, 1 \). Works for: \( n \ge 4 \implies n \ge 5 \). ## Even \( n \), Start With \( n \) Example: \( 6, 4, 2, 5, 3, 1 \). General: \( n, \dots, 2, (n - 1), \dots, 1 \). Works for: \( (n - 1) \ge 4 \implies n \ge 5 \implies n \ge 6 \). ## Even \( n \), Start With \( n - 1 \) Example: \( 5, 3, 1, 6, 4, 2 \). General: \( (n - 1), \dots, 1, n, \dots, 2 \). Works for: \( n \ge 3 \implies n \ge 4 \). ## Summary * Start with \( n \): Works for \( n \ge 5 \). * Start with \( n - 1 \): Works for \( n \ge 4 \).