Hex game and map coloring

I really enjoyed Camp Logic and am now reading another math book not geared toward teaching math. I am reading about the game of "Hex" created by Piet Hein. It has many versions and goes by other names as well. What I find interesting is that this book discusses how larger boards of Hex do not have known proven strategies (but on very small boards you can easily see how the first player can win with correct placement of their first piece). Winning strategies change with the shape of the board and I think this would be fun to explore in class along with the other strategy games in Camp Logic.

So my question is this... The book states that there is no formal proof or it's difficult to prove that the game cannot end in a draw. How can this be so? I think the book means that a mathmatical proof is difficult, but I think I can "see" a practical proof. If you try to fill a board up with two "colors" for two players, it is impossible to fill a board without creating chains. I'm looking at smaller boards and furthermore do not see a way to avoid forming a chain from one side to the other. It seems to me that you would need 4 colors/4 players to be able to fill the board without forming chains because each space touches 6 other spaces. Is this correct? Can it be related to map coloring? Can I show this in a class setting after playing the regular game? One neat variation of Hex is to force the other player to create a chain across the board. My students would be upper elementary/middle school.

The book is "Hexaflexagons, probability paradoxes, and the tower of Hanoi" by Martin Gardner. Thanks!