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abelian group additive group algebra answer is yes automorphism calculus Cantor function Cantor set Cesaro continuous Chapter checkerboard chord circles closed intervals colors commute complement Conclusion congruent Consequence consider convergent convex set countable set defined diagonal disjoint union dominoes easy elements equal equation exactly example exist extreme points factor FIGURE fixed point follows happens homeomorphic implies induction infinite instance intersection inverse isomorphic Lebesgue measure linear transformation look magic square mathematical matrix modulo monic polynomials monotone multiples non-empty open intervals plane polynomial positive integers positive measure positive number possible prime probability proof prove question rational numbers real numbers rectangle replaced result second player segment semigroup sequence of length Solution 12 square root square-free subset subspaces theorem tiled tion topological topological groups toss triangle trivial uncountable unit interval vector space words zero