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game with moving a button on a bar of 19 adjacent squares

Source: 2004 Estonia National Olympiad Final Round grade 12 p2

March 25, 2020
combinatoricsgamegame strategy

Problem Statement

Albert and Brita play a game with a bar of 1919 adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than 55, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the 1919th turn, Brita to be forced to move the button out of the bar.