A few weeks ago, Mark Steere emailed me with his latest game, Flume. If you are not familiar with Mark, he has designed a large quantity of combinatorial games, some of which take place on extremely creative boards!
Flume is played by placing colored stones on the intersections of a 6 x 6 grid (rules with pictures are available in this pdf). Before play starts, all intersections along the edge of the board are already covered by green stones that belong to neither player. The two players, Blue and Red take turns placing stones on the board. When you place a stone, however, you must look at the number of stones (of any color: green, blue or red) adjacent to your play. (Two stones can be adjacent only on the four cardinal directions, not on diagonals.)
This is where the game starts to look like Dots-and-Boxes: if your play is adjacent to three or four other stones, you get another play.
The game ends when the board is full; the winning player is the one who has played more stones. Since there are a total of 25 plays (5x5 intersections inside the green stones) there will never be a tie.
Mark gives some good examples in his rules explanation that show some cases where a player may not want to gobble up all the bonus-moves available that turn.
I got to play Flume here with a student last week and then again this week. Just yesterday, while playing, something exciting happened: the student realized she could use symmetric play to win when going first! The first move is to play on the center intersection. From then on, to counter the opponents' move, just make the same move reflected through that center point.
So, I would take the lower-right corner, and she would take the upper-left. I would play on the intersection just to the left of the center, and she would play just to the right.
This pattern worked well until I made an enticing move that opened up a free play. What is the best strategy here? Should she just play with the reflecting move? No, then I will take both of the bonus positions that turn, as well as any extra bonus moves created by them. After a little thinking, we realized the best solution: take the free space(s) and then make the same enticing move your opponent just made (unless you just ended the game by taking all the remaining positions).
Your opponent will gain (at most) the same number of "points" you just did. You will still have the upper hand since you went first and played the initial stone in the middle.
Of course, if the opponent decides not to take those bonus moves, then you can just claim them on your next turn.
The "tricky" part of this method (and the proof that it works) is putting your opponent in the same nice position they just put you in, and then showing that they won't be able to use those extra spaces to somehow jump ahead on their turn.
Naturally, this only works for play from the starting position and does not say anything about playing the game from an arbitrary position. There may be connections to Dots-and-Boxes there, but I don't know enough about the theory behind that game!
A Domino-Covering Problem
5 weeks ago