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This is Dan and Andrew’s Gaming Place.  We will be making posts on Tuesdays.


Announcing a book by Daniel Ashlock

bookpicOne of the research areas here at Dan and Andrew’s is figuring out how to get computers to produce game content more-or-less automatically. This post announces a book that summarizes many of our findings. You can buy a copy from the publisher Morgan and Claypool, but if you are part of a University or other institution that subscribes to the Morgan and Claypool synthesis series than an e-book with unlimited use for students and faculty will show up in your library presently. Profits go to Dan’s consulting company, which funds students and research!

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Filling a Rectangle with Elbows

An earlier post posed a problem of packing a rectangle with dominoes with the added condition that a packed rectangle contained no smaller rectangles that were also properly packed — that is to say any smaller rectangles inside the main one had a domino crossing their boundary. This post looks at the problem of this sort of “unbreakable” tilings using an elbow shaped polyomino with area three. Here is the simplest such unbreakable tiling.


What rectangles can be filled with the elbow shape at all?

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Heritage Universes

CoversA heritage universe is one in which there are incredible civilizations in the past and the most important feature of the world is the artifacts left by the earlier civilizations. There is a science fiction world called The Heritage Universe created by Charles Sheffield. The earlier civilization is called the builders, and they are known only by huge engineering works they left behind. Stories in the heritage universe are built around the artifacts left by the builders. There are several science fiction and fantasy universes that are built around the remains of older civilizations.

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Paracops, Shadow Hunters, and Men-in-black.

CFThis week Dan and Andrew look at the notion of outsiders — a group from outside a society, nation, or even world that maintain a presence for a benevolent or nefarious purpose. Our starting point is a notion from Charles Fort, an unabashed advocate of paranormal phenomena. Dan and Andrew find him and excellent source of ideas for role-playing games. He may have originated the idea we are calling “outsiders”, a term used very differently in D&D.

Or that there may be occult things, beings and events, and that also there may be something of the nature of an occult police force, which operates to divert human suspicions, and to supply explanations that are good enough for whatever,…
…or that, if there be occult mischief-makers and occult ravagers, they may be of a world also of other beings that are acting to check them, and to explain them, not benevolently, but to divert suspicion from themselves, because they, too, may be exploiting life on this earth, but in ways more subtle and in orderly or organized fashion.
–Charles Fort, “Lo!”

This notion of occult being from outside the current society or system actually translates into game mechanics in a number of ways.

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Recently, I attended a talk by Ryan Hayward, who is a Professor at the University of Alberta. He is publishing a book on Hex, and his research group has been working on how to play Hex for a long while. Hex is a two player game with simple rules, but it has more possible outcomes than a chess game!

The standard game of Hex is played on a board with hexagonal tiles, like the picture below.

The standard board is 11 x 11 hexagonal tiles, and it forms the shape of a rhombus. Note that the sides are coloured, so that opposite sides share the same colour.

How do you play Hex?

The thing I love about Hex is that is has very simple rules. On a given player’s turn, that player puts down a marker of their colour on an unoccupied space on the board. The first player to create a contiguous chain of markers from one of their coloured sides to the other side wins.

That’s it. Unlike chess or other board games, Hex is extremely simple to learn to play. However, things can get really complicated pretty quickly. It turns out that Hex is extremely hard to master.

There is also another really interesting property of Hex. In 1949, John Nash Jr. (made fairly famous by the movie A Beautiful Mind) proved that Hex cannot end in a tie, and that the first player can always win with optimal play. Unfortunately (or fortunately, depending on your standpoint) his proof was only about existence. He did not say anything about what the optimal strategy would be. So far, the game has been completely solved for everything up to 9 x 9 boards, but the standard 11 x 11 is simply too large at the moment for even very powerful computers. According to Wikipedia, in 11×11 Hex, there are approximately 2.4×1056 possible legal positions, this compares to 4.6×1046 legal positions in chess.

The other nice thing about Hex is that it is free to play, if you are happy to just print out a piece of paper with a board on it, and use some random tokens for each player, or even two colors of pen.  Here are printable boards of  size 9 and size 11.

If you are interested in other pencil-and-paper games, look at our sister blog Occupy Math.

Got anything more about Hex you want to add? Comment below! This is Andrew, from Dan and Andrew’s Game Place.

Dragons: what are they?

Long ago, when I was first learning about fantasy role playing, another player explained to me that the problem with dragons is that “they’re big and they die.” They also have a lot of treasure and so people always go after them. This was an AD&D campaign, and the description was basically correct. This campaign reacted to the problem by changing the number of hit points the dragons had into the number of hit dice they had. Dragons grew into demi-gods and the trouble needed to get their treasure was closer to balanced, but this seemed sort of excessive to me at the time. In this post we look at some other possibilities.

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The game of Goofspiel is a simple two player card game. Humans can play it pretty well but it is hard to get AI to learn to play — because the good strategy is obscure.  The game requires a standard deck of 52 cards, split up by suit. Player one gets all of the hearts, Player two gets all of the diamonds. The spades are shuffled and placed face down in between the players. The clubs are set aside. The cards are ranked as follows.  An ace is worth 1 point, the two is worth 2 points, …, the Jack is worth 11 points, the Queen is worth 12 points, and the King is worth 13 points.

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