No post: sick!

I've had a cold all week. More posts next week!

Nearly Shooting Myself in the Foot with Misere Sums

Oops!

Last week, in an effort to relive my success with finding negative games by summing to zero, I tried the same thing with a new game. The plan was to add states of a new game to states of games we had already covered, then see if they sum to zero. I wrote up states of the new game, and challenged students to find states in Clobber, Amazons, Nim, etc, that summed with the original game to get zero. Unfortunately, I hadn't had a good idea for a new game, so instead I decided we would play Misere Clobber.

Pretty quickly, students asked me: "Wait, how do we add a normal play game to a misere game?"

Oops!

Somehow I did some quick thinking (I usually can never seem to do this in front of a class) and reminded myself of how to make this "legitimate". I wound up writing two options on the board:

Either players are not allowed to make the last play in the misere game, or whenever a player makes the last move in the misere game, they immediately lose.

Whew! Things progressed pretty nicely at that point. Still, I was wary of teaching them that games can have the misere property instead of attaching that property to the method for playing a game. At least this took care of covering all the mechanics we needed to find negative games.

I took a number of wonderful pictures of the boards and of my students playing, but my new phone seems to have trouble with its camera and the pictures were never stored. Instead, enjoy these pictures one of my art-minded students drew on my whiteboard after we covered the definition of a game negative. :)

Searching for Games

There is no final exam in my combinatorial games class. Instead, the last four or so weeks will consist of student presentations. Each student is tasked with choosing a combinatorial game (that we haven't covered heavily in class) and researching "something interesting" about that game. Students will then present their findings.

The interesting thing does not need to be something super heavy, but should be non-trivial. So far I have three students who have chosen their games: Flume, Reversi and Hex. Of those three, two students have picked their "interesting things": one will code a playable version of Hex, while the other will describe the first-player winning strategy in Flume. In addition to these sort of options, students could write a program to determine the outcome class of their game, or just describe some interesting property (for example, that Hex cannot end in a tie or that the first player has a winning strategy in Chomp).

Part of my hope here is to learn more combinatorial games myself. I continue to work on expanding this table of games.

The other ten students have yet to choose a game. I have directed them to check out some resources such as Mark Steere's extensive list of creations, as well as the long appendix of games in our text, Lessons in Play.

Do you know of any other places I can point them to to find games? Certainly there is also a degree of procrastination, but it would also be great to give my students more resources. Also, it might come down to the point where I am forcing games upon the students. In that case, suggestions will be very helpful! Perhaps you've developed a game you'd like someone to check out!

Two turning points in my Games Class

Teaching this combinatorial games class has been tough. I am simultaneously teaching Software Engineering and Algorithms, and while both are challenging courses for students, they are going pretty smoothly. I know what I can expect from those students and know what I need to get across to them.

Combinatorial Games, on the other hand, is an adventure into some tricky territory. Since I have a wide variety of math and computer science students, I'm having a hard time making the correct assumptions about what my students already know. Since this is a new-fangled elective, I also don't have specific goals I need to communicate.

The structure of the course has been to spend one of the two class periods each week focusing on playing a new game. (Here is the class schedule so far.) The first week we played Domineering, then Clobber, then Toppling Dominoes. Students would play a bit amongst themselves, and do a great job answering questions I posed. But, after a few games and a few opponent-switches, they would get a bit bored. The 1.5 hour class started to drag on and a few students would actually ask me to return to lecturing. (Luckily, I brought some notes with me!)

Then, in the fourth week, we played Amazons. Wow. The students responded to this by actively getting into the game and trying to figure out good moves. The idea of outcome classes started to click and when I called for a change of opponents, very few people got up in the first minute.

Last week, we did something even better. I had the students play game sums (Note to me: I should have done this sooner!) and try to find different games that summed to zero. The base game was Amazons, and I drew different instances on the whiteboard and challenged the students to find different Domineering, Clobber and Toppling Dominoes games that, when added to the Amazons board, summed to zero. The result was possibly the best class period I have ever taught... even though I personally did very little. Students quickly took to their game boards, reasoned about some sums, then started filling up the marker board. Almost immediately some things written on the board were challenged (though no one was brash enough to erase another student's work without permission) and some excellent discussion began to take place.

Wow.

It's hard to describe that level of engagement by students. Everyone was knee-deep in advanced mathematical material, experiencing it first-hand. We haven't yet defined many possible game values (I'm not sure we've defined anything rigorously yet) but students were quickly clamoring for an explanation of different fuzzy games and non-number values.

As I continue this semester, I think I need to make sure that every topic is motivated, and perhaps play games in each class (instead of every other). These last two game days have made an amazing argument for that!

Out again tomorrow

Unfortunately I will be out again tomorrow. With any luck I'll be back in action next week.

Quantum Chess

Recently there was some cool board game buzz about combining quantum super-positions and chess: Quantum Chess.

The game was designed by Selim Akl as a response to the brute-force superiority of computers in standard chess. Alice Wismath, working with Dr. Akl, implemented a non-quantum version (they point out: "a true quantum board may be a few years in the future") as a Java 1.6 applet (I had to upgrade my browser's Java).

The basic idea behind this game is that the identity of each piece (aside from Kings) exists in a super-position before it is moved. The actual piece will be one of two different options (for example, either a rook or a knight) which is only known once you decide to move that piece. Thus, each turn consists of first choosing a piece to move, then determining what type of piece it actually is, then moving that piece.

Naturally, since the value of the pieces is based on some randomness (quantumness is considered randomness, right?) this is not strictly a combinatorial game. Until we have quantum boards, it's not exactly a board game either... Still, we can implement this in a non-quantum way using a big checkerboard and two sets of chess pieces. By putting two pieces on the same square to indicate the super-position for non-collapsed pieces, you can then decide the actual value by flipping a coin once the piece is chosen.

In any case, this is an extremely original game and an excellent work by Akl and Wismath. With any luck this will bring interest into both games and general quantum... ness. As you can see, I need a lesson on quantum mechanics and quantum computing!

Note: I will be out of action on Tuesday, so the next post will probably not occur until next Friday.

Oops!

Hmmm, apparently I've been a bit confused recently about getting posts out at the correct time. I somehow posted twice last week on Tuesday without realizing it until Friday.

Hmmm...

Unfortunately, I have just run out of time today.

I will have something new to say on Friday and will get back on my regular schedule.

Sorry!