Game Description: Clobbineering

The first time I taught a games class, my student Will Herrmann combined the games Clobber and Domineering into the excellent ruleset Clobbineering.

If you're familiar with both of these games, perhaps the rules are already clear to you. In case they're not, a turn consists of either making a legal clobber move with the checkers on the board or placing a domino on two empty spaces a la domineering. Thus, the player that clobbers with the black checkers is playing dominoes vertically, while the red-or-white checker clobber-player plays dominoes horizontally.

This is enough to make the game states quite difficult to analyze. Often times the game will seem like it's over in one player's favor, but then there's a really sneaky move that changes everything.

My usual strategy is to concentrate on clobbering to open up domineering plays for myself for later that don't allow my opponent to play dominoes. I don't have any better considerations than that; if my opponent catches the drift, I'm generally in trouble!

Game Sum Demonstration Videos

A few weeks back, my aide, Ernie, and I played some game sums as demonstrations. Of special note, you may not have agreed with the non-all-smalling of Hex, and that may make you want to watch the last three!

Here are the links to videos:

Domineering + Clobber [6:32]

Domineering + Checkers (Draughts) [5:24] (Warning: this one is unsatisfying!)

Domineering + NoGo [4:14]

Y + NoGo [7:32] (Using the non-all-small Y)

Hex + NoGo [4:27] (Using the non-all-small Hex)

Hex + Clobber [13:16] (Using the all-small Hex!)

Clearly Combinatorializing Connection Games: Un-all-smalling

Game sums are at the heart of Combinatorial Game Theory. If you give me two different rulesets, a third exists that is the sum of those two and you don't have to specify anything new.

Unfortunately, some games add poorly or uninterestingly. Hex may be a good example, because the game is over when a path is created. The standard Hex rules cause the game to be all-small: if one player has a move option, then the other also does. As I mentioned before (late in the post) it's perhaps more exciting to redefine the game to make it more "sum-friendly" (very arguable). Instead of ending the game when one player has created a path, instead allow the path-creating player to keep painting uncolored hexagons. Thus, the rule is that you cannot play if the opposing color has formed a path. Now the game is no longer all-small.

Paul Ottaway and I had a conversation a year ago where we both argued for this change. I don't know how to change the rules to a 65-year old game, however. Luckily, when played atomically (not part of a sum), the new rules do not change the game play. Make the connecting path and you win.

This works for any connection game that I know of (Y, Twixt, etc). There are many inherently all-small games that cannot accommodate such a change, however. Clobber and MadRooks are games that are all-small and for which I don't see a method to fix that that doesn't change the original game.

The inherent problem here is that prior to combinatorial game theory, games were defined by explaining the conditions for one player to win the game. Under the normal play convention, that's not entirely relevant. Instead, the game author need only describe what the legal plays are for each player from any position. It's a subtle difference that is only important in the context of game sums.

Game Description: Mad Rooks

This semester I am lucky enough to be working with a student on an implementation of Mad Rooks for the Android OS. In a previous post, I talked a bunch about the length of a Mad Rooks game, but I realized I didn't have a stand-alone post on the game itself.

Mad Rooks is another game developed by Mark Steere who has created a large number of excellent combinatorial games. This particular game is a sort of King-of-the-Hill Long-Distance Clobber. Both players begin with pieces that act like rooks in Chess in that they can move as many spaces horizontally or vertically on a checker board without jumping pieces. If a rook moves onto a space occupied by an opponent, the opponent's piece is removed (captured). A rook is called "engaged" if it can make a capturing move. Each turn, a player either uses one of their rooks to capture a piece or moves one of their unengaged rooks so that it is engaged.

Just as with Clobber, this game is all-small, which means if there is a move for one player, then both players have a move. Although this does not mean that each game has a nimber value (though it always seems like it should to me) it does mean that every instance of an all-small games has an infinitesimal value. Infinitesimals are smaller than all positive numbers, yet greater than all negative numbers (the only infinitesimal number is 0). Values such as Up, and * are infinitesimals.

It is especially interesting to look at end states in Mad Rooks, where suddenly making engaging moves instead of capturing with another piece is a better move, though engaging when you have to is often disastrous. It doesn't always even matter who has more pieces. Consider a board with rooks only on every square along the diagonal of the checkerboard. Now, if only one of those is Blue but the rest are Red, the next person to play loses the game, despite the fact that Red outnumbers Blue seven-to-one! (Try it!)

One reason some players may prefer this game to Clobber is that in the end all pieces of one player are captured, leaving the winner as the last color standing. In Clobber, you can't look at the final position and determine who won the game, but you can in Mad Rooks!

Game Description: Martian Chess

Just last week, I forced my aide to sit down and play Martian Chess with me. This is one of the games I was introduced to at Origins this past year, and have been looking forward to the start of the semester to try it out more.

This game is sort of a more complex version of Clobber: pieces are arranged on a checkerboard, and they "clobber" pieces near them---except that there are 3 different types of pieces and they move in different ways. Pawns move one space in any direction (including diagonal), drones may move up to two spaces (but only horizontally and vertically) and queens may move exactly as queens in Chess. Pieces may not jump over other pieces. Okay, maybe it is more like Chess than Clobber...

Nevertheless, the goal is to take opponents' pieces. 3 points are given for a queen, 2 for each drone and 1 for each pawn captured. The game ends when a player no longer controls any pieces. The twist, however, is that when you capture an opponents' piece, they now take control of your piece on the board. This occurs because each player "owns" two opposite quadrants of the board: they may move any pieces in their quadrant. Capturing an opponents' piece means moving into an opponents' section and surrendering the piece you just owned.

Martian Chess is a really fun game that forces you to think on your toes. Just about when I seemed to be coming up with a strong strategy, Ernie pulled a new trick on me and had me completely second-guessing myself. I would really like to play this game in class, but, alas, I don't think I have enough Icehouse pieces!

Let's use this wacky ownership in a variant of Clobber: Reverse Clobber. Now, when I clobber an opposing piece, I instead lose my own piece. How much fun is this to play? (Maybe not so much...)