An AbstractStrategyGame (actually called Connect6 but that's not a WikiWord) invented by computer science Professor I-Chen Wu of National Chiao Tung University in 2003.  He supposedly invented the game while playing GoMoku with his daughter.

In this game, two players, Black and White, alternately place two stones of their own colour, black and white respectively, on empty intersections of a Go-like board, except for that Black (the first player) places one stone only for the first move. The one who gets six or more stones in a row (horizontally, vertically or diagonally) first wins the game.

The fact that every move after the first move of one stone is two stones makes the game very fair, because at the beginning of a move the player moving will have one less stone than the opponent and at the end of said move the player moving will have one more stone than the opponent. Contrast this with Go, Hex, or Gomoku where the player who moves first will never have less pieces on the board than his opponent and therefore gains a very large advantage.  For example, in Gomoku, the game was still a win for Black(who moves first) even with restrictions put on Black's moves(the game with restrictions is called Renju).  However, the top Renju players who have examined and played Connect6 still do not know which player, if any, the game favors.

Can be played on LittleGolem at http://www.littlegolem.net/ as well as many other game.