Changing the series of natural numbers
Source: Russian TST 2022, Day 6 P3
March 21, 2023
combinatorics
Problem Statement
Write the natural numbers from left to right in ascending order. Every minute, we perform an operation. After minutes, we divide the entire available series into consecutive blocks of numbers. We leave the first block unchanged and in each of the other blocks we move all the numbers except the first one one place to the left, and move the first one to the end of the block. Prove that throughout the process, each natural number will only move a finite number of times.