## Friday, February 18, 2011

### Game Description: Weak Hex (Whex)

When I was first interested in Adjacent Hex (Adjex) a few years back, a friend of mine pointed out another similar game to me: Weak Hex. Weak Hex, or Whex, is extremely similar to adjacent hex because both players must move adjacent to another player's move. Instead of playing next to the last move of either player, however, both must play adjacent to the last move of the red player!

Here is the description from the original Whex paper by Argimiro A. Arratia Quesada:
Players can not colour an arbitrary vertex but must proceed as follows: Player
1 begins the game and he must do it by colouring red a vertex adjacent to
the source. From this move and on, Player 2 must colour blue an uncoloured
vertex adjacent to the vertex last coloured red (i.e. coloured by Player 1), and
Player 1 replies by colouring red an uncoloured vertex adjacent to the vertex
that he coloured red last.
(If no adjacent moves are available, the next player immediately loses.)

This game is intended for play on a graph instead of the standard hex board, but we can consider that also. In this case, the "source" is likely to be an entire red side (the sink, then, is the other red side). The goal for the red player is to connect the two red sides, and the goal for the blue player is to prevent such a connection. This matches up precisely with the winning conditions of standard Hex, so it is possible to play Whex on the regular hexagonal grid.

Additionally, the game is shown to be PSPACE-complete, so it is certainly hard to play in the general case! This is the same as with Hex and Adjex. In Hex, it is known that the first player has a winning strategy, but that strategy itself is not known. In Whex, however, not only does the first player have a winning strategy, but it can be described very quickly! In fact, I bet you can come up with it yourself!

(Find an opponent, see if you can come up with a method to always win playing first (as red), then come back and finish reading!)

Okay, on your first turn, play in the hexagon "on the bottom"

On each subsequent turn, play in one of the hexagons directly towards the upper-right-hand (red) side. Blue can only choose to occupy (at most) one of those on their turn, so you will get to take the other. Since you don't have to worry about playing adjacent to them, they can't deter you (as is possible in Adjex). Enjoy your march to victory!

If you play with the pie rule, I'm not sure what happens!