First, a few administrative notes. Wittenberg's classes end on Wednesday, so this will be the last Friday post of the semester, and Tuesday will end the regular schedule. I have a list of potential topics on my whiteboard for Tuesday, but if there's something you'd like me to comment on, please let me know.

This summer I will be away from Internet access a great deal and I will not try to keep up any sort of scheduled postings. I'm sure there will be some posts, but not with any regularity.

Thanks also to everyone who has posted or emailed me about content this year! One of these people was Paul Ottaway, who suggested I try playing the game Fjords. He mentioned that this is half non-combinatorial and half combinatorial.

Earlier this month, I got a copy for my birthday and I've already played a bunch of games! It is just as Paul mentioned: the first stage (which actually takes a long time) uses a lot of random elements. The second stage (this goes pretty quick) is a pure combinatorial game. It is not, however, a game that I am aware of. Perhaps it exists and has already been studied! Perhaps you will have heard of it and can let me know! :)

The first stage of the game consists of the players "exploring" the land they will settle. Players flip over hexagons with field, mountain and sea patterns, fitting them together to form the landmass. The result of this is a hexagonal grid graph with some vertices missing (tiles were not placed or do not include any field area) or edges missing (field tiles with mountains or the sea between them are not adjacent). While placing these tiles, a player may elect to place one of their few farms on the most recent tile (it's stuck there for the rest of the game). Thus, the hexagonal graph has some of its vertices labelled either Red or bLue before the second stage.

The second stage of the game is then very simple: a players' turn consists of labelling a vertex. That vertex must be both uncolored and adjacent to a vertex already of that player's color. When one player cannot color a vertex, they lose. In the actual game, these newly colored tiles represent fields spreading from your farms. Also in the actual game, if you both get the same number of farms, it is a tie (instead of a second-player win).

The second half of this game seems very basic, however. I would be astonished if it didn't have a name in combinatorial games. Even if played on any (planar?) graph instead of only a subgraph of a hexagonal grid, this must be studied somewhere.

In any case, I highly suggest giving Fjords a try! It's an excellent game for two people, but do not believe the 30 minute time requirement the box suggests (they want you to play the whole thing three times). All my games take around 30-45 minutes each, meaning a WHOLE game would take around 2 hours!

A Domino-Covering Problem

3 months ago