Playing without Giving away your Strategy

Perhaps you are about to play a series of three games of cram against an opponent in the final round of a cram tournament. Your scored better than your opponent in the tournament thus far and must play first in the first and third games. The winner of the tournament will be the player that wins 2 of those 3 games.

While playing the first game, you can see that you are losing, but suddenly realize there is an really good strategy for winning from the start as long as you go second. Since this tournament uses initial 8 x 8 boards, you can use a symmetry strategy to mimic the other player's moves! Great! The next game is yours!

Unfortunately, you also realize the strategy is easy to follow. As soon as you use it against your opponent, they will be able to implement it themselves in the final game and win.

Is there a way to implement the symmetric-play strategy but disguise it at all? In general, is there a way to mask the fact that you are using a simple strategy to win?

In Nim, it is difficult to follow someone making the correct moves if you do not know how to evaluate the game boards. If it was plainly obvious what the other player was doing (perhaps XOR is your favorite math operator) then perhaps you would pick up on it quickly.

I tried this once, keeping a "turn behind" in the symmetric-play strategy. I kept making the play my opponent had made two turns ago, whenver possible. As I recall, it got messed up and I had to ditch it part way through. Is there any good way to keep this going? I feel this would be more difficult for an opponent to follow... if it works!

Game Description: Domineering

First, thanks to people who suggested I add rows to the game table page. Please suggest more games to add for rows!

In the past two weeks, I have gotten more chances to play games with Wittenberg students over lunch. This turns out to be one of the best ways to try out new games! Exploring them with other newbies is very helpful and the right attitude can add a non-competitive spin to the discovery of good strategies. Unfortunately, since these students have not been studying combinatorial games, I usually wind up with an unfair advantage. Not always, though. This is the best: getting beaten by missing a great strategy that gets pointed out by someone else! Seeing the realization of that great move in a student as they make it is wonderful!

Unfortunately, next I have to admit that I had never played Domineering before. Domineering is the guiding example for many basic combinatorial game aspects, yet I had never pulled out the dominoes and checkerboard before with another player. Actually, I don't even own dominoes; I had to borrow the set from our Math lounge.

Domineering is a partisan game between two players where each turn a domino is placed over two adjacent (share a side, not a corner) squares. The Left player must place dominoes vertically while Right must place them horizontally. You may not place dominoes on squares already occupied by a domino. Assuming normal play, the first player who cannot place a domino loses.

Domineering is a wonderful example for combinatorial games and game positions. It is used as the first guiding example in Lessons in Play, mostly because simple game values can be easily found. Good moves are sometimes very easy to spot, even long before the game ends. Perhaps most important, as the game continues, it clearly decomposes into smaller subgames that add together.

Still, while playing with students lately, one of them quickly became bored. "I wish we could play the pieces either way," she said (or something like that).

"That game's called Cram," I said. "I've also never played that one. Wanna give it a go?"

We played Cram, then quickly analyzed end game scenarios. We played it a bunch more times, allowing backtracking near the end and generally figuring out how a lot of the game worked. The next time, we skipped the Domineering and went straight to Cram.

This contradicted my assumption that many people prefer proper partisan games over impartial games. Is there a general consensus between these two games?