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2 player game with 2n-digit num divisible by 9

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

November 21, 2022
number theorycombinatoricsgamewinning strategy

Problem Statement

Let nn be a positive integer and M={1,2,3,4,5,6}M = \{1, 2, 3, 4, 5, 6\}. Two persons AA and BB play in the following Way: AA writes down a digit from MM, BB appends a digit from MM, and so it becomes alternately one digit from MM is appended until the 2n2n-digit decimal representation of a number has been created. If this number is divisible by 99, BB wins, otherwise AA wins. For which nn can AA and for which nn can BB force the win?
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