Presenting Neighboring Nim to Undergrads

I replaced getting a post ready for Tuesday with preparing a talk for this past Monday. I spoke at our department colloquium about Neighboring Nim, describing the reduction from (Vertex) Geography. Two excellent things happened.

First, I had a great audience. While playing Neighboring Nim against them, they were very energetic to make good plays against me. Will, a senior student of mine, immediately took on the role of surveyor, collecting move nominations from the crowd, then calling for a vote. The discussions were loud and boisterous. At one point, I made a very fast response to one of the audience's moves, using the Quick and Confident method. Unfortunately, I moved to an N position, with two P options. The audience quickly picked the board apart (it was by no means trivial) and found the two winning moves. The problem was that, individually, they didn't see the viability of the other move. So there were two groups arguing vehemently for two separate perfect moves. The attitude nearly boiled over. Luckily, the audience finally agreed and they made one of the moves. It was very exciting to speak to such a charged group!

Second, I changed the order of my talk around regarding the proof of computational complexity. Usually I claim that a game is hard for some complexity class, explain what that means, then show the reduction. Talking about complexity classes to a general undergraduate audience is a bit of a speed bump, and often shakes people out of the rest of the talk, meaning they don't follow the steps of the reduction. This time, I just said that the two games were related, then explained the reduction. This kept the crowd interested, because it's a description of positions in a game they just learned how to play. At the end of the talk, I spoke briefly about the implications of the transformation: the PSPACE-hardness of Geography implies the PSPACE-hardness of Neighboring Nim. At that point, having seen the reduction in full, I hope it sank in. Although I wouldn't do this in the future when presenting to a primarily graduate-student/professor group, it seemed to work really well for people who hadn't seen computational complexity before.

The slides for the talk are here.

Next week is Fall Break at Wittenberg, so there will probably not be a post on Tuesday as I spend the time off preparing for Integers 2011 the week after. With any luck, once I'm there I'll get to post some notes each day.

Game Descriptions: NimG and Neighboring Nim

What happens if you restrict which games can be moved on in a game sum? This seems to be the idea behind NimG, a game played on a collection of Nim Heaps embedded in a graph. There are a few variants, but each turn a player does two things:

* traverse an edge of the graph adjacent to the last move, and
* remove sticks on the nim heap embedded either into the edge traversed or one of the vertices.

In the sticks-on-edges version, players remove sticks from the edges while traversing them during the turn. Naturally, if the heap on the edge is empty, it cannot be traversed. In other words, the player who starts their turn on a vertex where all incident edges have empty nim heaps loses. Masahiko Fukuyama studied this game a bunch and more recently Lindsay Erickson has looked into instances on the complete graph.

When sticks are embedded into the vertices, a move still consists of traversing one edge to arrive at a new vertex. Upon arriving at the new vertex, a move is made on the nim heap embedded there. Gwendolyn Stockman may have been the first person to analyze this game in an REU advised by Alan Frieze and Juan Vera. This is the version that interests me: the vertices act like different heaps in a game sum, but the edges restrict which heap can be chosen each turn. In this ruleset, the complete-graph version is equivalent to standard Nim. (If each vertex is adjacent to a "self-loop", an edge connected twice to the same vertex.) In this game, you lose if all vertices adjacent to the current location contain empty nim-heaps.

In order to avoid the self-loop issue, I altered the rules slightly and set out to do some analysis. Neighboring Nim is exactly the same game as (Vertex) NimG, but a player may choose not to traverse an edge and instead remove sticks from the current vertex (again).

I have actually played this game very rarely, mostly because I can't imagine a standard starting position. Nevertheless, I helped show that the game is PSPACE-hard. I was lucky enough to present this result at BIRS in January, which was quite pleasant. The paper should appear in Games of No Chance 5. (arXiV version)

The cool thing about the computational complexity of this game is that once all vertices have maximum heaps of size 1, the game is the same as undirected vertex geography, which is solvable efficiently. In order to get PSPACE-hard instances, we needed most vertices to have a heap of size 1, with only a few having a second stick on them (less than 1 in 7 of the vertices need the extra stick)! It's very interesting that such a small rule change can result in such a big complexity change!