An Inductive Proof of Hex Uniqueness

A short, inductive proof is presented of the fact that a Hex board cannot be colored such that winning conditions are satisfied for both players. It is well known that the game of Hex, independently invented by Piet Hein and John Nash, always has a winner. In it, two players, Black and White, attempt to connect opposite sides—East and West or North and South—of a parallelogram tiled with hexagons by coloring tiles with their respective colors (see Figure 1). We call these sides for the respective players necessary edges. Similar to this result is the intuitively obvious fact that the board cannot be colored such that there are two winners. In 1979, David Gale mentioned a proof of this by “induction on the size of the board,” but did not present it [4, p. 820]. In fact, no such inductive proof has, to our knowledge, ever been published.