  In Problem #1542 of Mathematics Magazine, Grossman and Turett define the
Cantor game. In his 2007 Mathematics Magazine article about the Cantor game,
Matt Baker proves several results and poses three challenging questions about
it:
  Do there exist uncountable subsets of [0, 1] for which: Alice does not have a
winning strategy; Bob has a winning strategy; neither Alice nor Bob has a
winning strategy?
  In this paper we show that the answers to these questions depend upon which
axioms of set theory are assumed. Specifically, if we assume the Axiom of
Determinacy in addition to the Zermelo-Fraenkel axioms, then the answer to all
three questions is "no." If instead we assume the Zermelo-Fraenkel axioms
together with the Axiom of Choice, then the answer to questions 1 and 3 is
"yes," and the answer to question 2 is likely to be "no."
  Author's Note: This paper was my entry in the 2017 Regeneron Science Talent
Search. It earned a Top 300 Scholar Award as well as Research Report and
Student Initiative badges.
