We started with Hex. It is a game which is easy to describe but not necessarily easy to play. It was invented by the Danish mathematician Piet Hein in 1942. It was independently invented by mathematician John Nash in 1947 at Princeton University. The board is a rhombus shape. Players must connect two opposite sides with pieces of their colour. There is a mathematical proof that this game cannot end in a draw. There are lots of resources online describing the strategy. You can play Hex online.
It is very difficult to buy this game so I printed paper boards. This might have been one reason why the children were reluctant to play it for too long. We moved on to Bridg-It (Hasbro, 1960). This variant of Hex was invented by mathematician David Gale. Players attempt to build a continuous connected bridge of their colour from one side of the board to the opposite side, while blocking their opponent from doing the same. When testing this game I could see the similarity to Hex and wondered if a game could ever end in draw. I searched online and found that the only way a bridge can be completely blocked is by completing a bridge of the opposite colour: draws are therefore impossible. You can play Bridg-It online.
Our final game was Quoridor (Gigamic, 1997). The aim is to be the first player to reach the opposite side of the board. On each move a player has two options: moving their pawn one square, or placing a fence. If you place a fence you can obstruct the other player and slow them down, but if you are not careful you can end up obstructing yourself. I will write more about this game when I have further explored games on the theme of mazes and labyrinths.