and the Brouwer fixed-point theorem can be proved from the Hex theorem: "every game of Hex has at least one winner", from which we obtain a logical implication: Hex theorem implies Brouwer fixed-point theorem, which implies Jordan curve theorem.

It is clear that Jordan curve theorem implies the "strong Hex theorem": "every game of Hex ends with exactly one winner, with no possibility of both sides losing or both sides winning"; thus, the Jordan curve theorem is equivalent to the strong Hex