History of the Game

In the 1940s, two mathematicians (Piet Hein and John Nash, the Nobel-prize winner made famous in “A Beautiful Mind”) independently designed a game now known as Hex. It is played with stones of two colors on a board pictured at right.

Each player owns all the stones of one color, and the players take turns placing stones, one by one, on the board. The object is to construct a chain of stones in your color that runs between two opposite sides of the board. Your opponent tries to construct a chain of his stones running between the other two sides of the board. The second picture at right is an example of a finished game, won by red.

Hex is a landmark in game design: it has simple rules, deeply strategic play (at least one book has been written on Hex strategy), and some attractive mathematical properties.

In Hex, each player tries to form a particular kind of pattern. Its beauty lies in the fact that the patterns are mutually exclusive, but one is inevitable: when the board is full of stones, one and only one of the patterns must be formed. Thus, neither draws nor infinite positional repetitions can occur.

hex1 Hex2

Beyond these beautiful and convenient properties, Hex has a special relationship to the human visual system. The brain is evolved to detect patterns, but it's better at detecting some kind of patterns than others. It's especially good at grouping similar elements together in order to see "higher level" objects. For example in the pictures above, you can see the board not only as individual cells of color, but as "islands" of color, where each island is made up of several like-colored cells. This capacity for grouping allows the human brain to play Hex and games like it intuitively and at an extremely high level (with practice, of course!). To wit: while the best Chess programs are now better than even world champion chess players, nobody has yet figured out how to program a computer to play Hex well against good human opponents. It's not because computers are stupid, but rather it's because we are geniuses at this particular kind of game. At the moment, no one can figure how our brains do it, but they do. As a neurobiologist and mathematician, I find this fascinating.

In 2006, I asked myself: can I construct a game like Hex, having the same nice mathematical properties, but where the players themselves determine the particular patterns needed to win, so that the patterns can change from game to game? I found that the answer was yes. In order to understand how, one must first see that there is another way to describe the goals of Hex.

The traditional way to teach Hex is to describe to each player the pattern that he must try to form, but it’s not the only way. Instead, you can tell one player to try to form his pattern, and then tell the other player to try to block it. You only need to describe one of the patterns, and the other will emerge naturally when one player tries to block the described pattern.

With that in mind, I realized that the pattern you describe doesn’t have to be a chain between opposite sides of the board. It could be anything. If you specify almost any pattern, there will be another one that can block it, but you needn't be able to describe it in order to play.

That was the key insight, but I didn't have a game yet. In order to make a fair game, one must ensure that the goals of the two players are about equally difficult. Let's call the two players the builder and the blocker. In order to make the game interesting, the builder and blocker must have equal chances, if their skill levels are equal. There are lots of patterns that are easier to build than to block or vice versa, so whoever decides the pattern can easily win if he knows whether he’s the builder or blocker beforehand.

I found a simple solution: one player should invent the pattern first, and only after that should the other player decide who is builder and who blocker. That way, the player who invents the pattern must choose a balanced one, or else his opponent will get an advantage by taking the easier role. With this rule in place I had my first version of the game. The “Rules for Casual Play” are nearly identical to that original version (the only difference being that I’ve added an additional method for balancing the players’ goals).

The “Rules for Tournament Play” solve another problem that I discovered later, when I tried to imagine what would happen if players studied the game intensively, like some study chess. I realized that a player could study intensively one particular pattern, so that he knew more about it than others did. Then, in any game in which he had the opportunity to propose a pattern, he could propose that one. Even if the pattern was balanced, his expertise in that pattern would grant him an advantage. To fix this, I added a bidding phase at the start of the game, where players bid for the right to propose a pattern. With that, the game was done.

So, Mind Ninja is a generalization of the rules of Hex to any arbitrary pattern goal. But it’s not only a generalization of Hex. Many other games, Like Y, Star, 5-In-a-Row, and One-Capture-Go, as well as many others, are all contained as specific patterns in Mind Ninja.