Weekly "Game Lunch"

This semester, I've (finally) gotten a social game lunch at Wittenberg in full swing. This is a great way for faculty, staff and students to interact casually, but also exercise our minds!

I've had a couple past semesters that were attended regularly, but now it is finally common to have different groups of people wanting to play different games on any given week.

Some aspects that seem to work well are the following:

* Short games with variable numbers of players. Tsuro, Tetris Link, World War 5, and Hey, that's my Fish! have all been popular.

* Games with less hidden knowledge tend to promote discussion. The less time you spend not on your turn trying to figure out what you're going to do next and more time paying attention to what other people are doing, the more you may interact with them.

* Games with shorter turns keep people interested. Hey, that's my Fish! has a problem where players may try to analyze the game too deeply to try to win this turn. This can drag turns out unnecessarily. This relates to the next point...

* Games with randomness seem to keep turns shorter. Although they're no longer combinatorial, there's still plenty of discussion/consideration in these games to keep them interesting. I bet some will disagree with me here...

I fully recommend adopting a weekly game lunch! :) Our Tuesdays have become very fun! So much fun that I've missed a bunch of Tuesday posts lately! Oops!

Next week is Wittenberg's spring break. Posts will return the following week.

Game Description: Rule 60 Game

Urban Larsson gave a cool talk at Integers 2011 about combinatorial games based on cellular automata.

Unlike the Game of Life (which is an automata, but not a game in the "combinatorial" sense), Urban concocted actual two-player impartial rulesets based on Wolfram's rules 60 and 110. I'm not familiar with these, but I do think the two games are very interesting!

The Rule 60 Game is a ruleset that uses a pile of matches and a pile of tokens. On turn i, the current player may remove as many matches as they want on their turn (at least 1) and a number of tokens between 0 and the number of matches taken last turn (inclusive). Additionally, you may not take the last match unless you also remove the last token.

Consider the position with a heap of four tokens, a heap of two matches and that last turn there were three matches removed. What is the outcome class of this game?

As it turns out, this game is in Fuzzy. You can't end the game this turn by taking both matches) because you can't take all of the tokens. Thus, the current player has to take one match and either zero, one, two or three tokens. By taking three tokens, the resulting game is still in fuzzy, as the next player wins by taking both the last match and token. Taking zero, one or two tokens, however, leaves the opponent unable to move because they cannot legally take the last match. Any of these three positions are in Zero.

Urban has also created a game based on Automata rule 110. Although the relevance to automata-theory is a bit lost on me, these games are interesting nonetheless!

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 Description: Odd Scoring

Oops, I previously talked about Odd Scoring without giving a descriptive post. Let me fix that now!

One of my students presented this game in my first-year seminar to the class last semester. Here are the rules.

There are a total of an even number of horizontally-set spaces, with one marker placed at the far-right space. Both players keep their own score, which starts off at 0. Each turn, a player moves the marker 1, 2 or 3 spaces to the left, then adds to their score that number of spaces moved. Once the marker is in the far-left space, the player with an odd score wins.

Some interesting points:

* exactly one player will have an odd score, because there are an odd number of moves the marker will make in total.
* you really only need to keep track of the parity of the players' score. Having 9 and 3 points are the same.
* this is not strictly a combinatorial game; the last player to move may not be the winner!

Even though this game is not quite combinatorial, can it be considered all-small? Whenever one player has a move, the other also has a move. However, in some cases this is a move where the game ends, but the last player to move still loses. Consider the following game state:

_X_ ___ ___ ___ ___ ...

Left: 0 Right: 1

What is the value of this position? That definition may change whether this game is considered all-small or not! This may not be consistent with the definition of "points" considered earlier this week!

Combinatorializing Games: How do points actually work?

We've talked recently about combinatorializing connection games, but what about point-based games. Here's a general definition of a point-based game:

Players make moves and earn points during the game. This continues until there are no possible moves. When all moves have been made, the player with the most points wins.

What makes this not-quite-combinatorial is that it is not necessarily the case that the last player to move is the one who wins the game.

Flume is an example of such a game. Players score "a point" for each piece they play, and at the end the player with the most of their pieces on the board wins. Since there are an odd number of spaces, there will be no tie.

What happens, then, if you add two games of Flume together? What if you add a game of Flume to a game of Hex?

The way I've always envisioned these games working is as follows. Let's say the game ends with the left and right players each with their point totals (called left-points and right-points, respectively). Then a new game immediately starts with value: left-points - right-points, the winner of which wins the whole thing.

Thus, if you play a game of Flume and at the end the left player (Blue) played 9 disks while the right player (Red) played 16, then the result is game of value: -7, which the right player should win.

Is this how the "combinatorialization" is usually conceived? Is there another good way to handle this?

EDIT: Fixed a typo in the title. (Feb. 10, 2012)

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.