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2 player game, removing 512, 256, 128, ..., 1 numbers from {0,1,..., 1024}

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

November 22, 2022
combinatoricsnumber theory

Problem Statement

Two people AA and BB play the following game: They take from {0,1,2,3,...,1024}\{0, 1, 2, 3,..., 1024\} alternately 512512, 256256, 128128, 6464, 3232, 1616, 88, 44, 22, 11, numbers away where AA first removes 512512 numbers, BB removes 256256 numbers etc. Two numbers a,ba, b remain (a<ba < b). BB pays AA the amount bāˆ’ab - a. AA would like to win as much as possible, BB would like to lose as little as possible. What profit does AA make if does every player play optimally according to their goals? The result must be justified.
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