Mind Ninja The Game of Boundless Forms

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1 Mind Ninja The Game of Boundless Forms Nick Bentley Overview Mind Ninja is a deep board game for two players. It is 2007 winner of the prestigious international board game design competition Concours International de Créateurs de Jeux de Société. Objective Board Initial Steps 1. Bidding 2. Choosing a goal pattern 3. Setting initial position 4. Choosing a builder and a blocker Play End of Game Patterns History of the game Objective In the game, a player choose a goal pattern, and then one player tries to build it by placing colored stones on the board while the other player places stones to prevent the pattern from being built. Board In general the game can be played on a board of any size and shape. There are two standard tournament boards, however: a hex board and a square board, both of which can be seen in the pattern section. Initial Steps. 1. Bidding 2. Choosing a goal pattern 3. Setting initial position 4. Choosing a builder and a blocker

2 1. Bidding (Note: this step is optional, and should only be used when playing competitively or in a tournament setting, where a game clock is available. If playing with a game clock, each player should start with 25 minutes on his clock) The purpose of this step is to determine which player will get the right to choose the goal pattern. This right is given to a player who is ready to give up more time on his clock than his opponent. Both players begin by bidding numbers which stand for time. The players alternately increase the bid by one minute until one player passes. The player who did not pass then reduces the time on his clock by the amount of his last bid. 2. Choosing a goal pattern The player who did not pass in step 1 above chooses a goal pattern from among the list of patterns at the end of this document, and shows it to his opponent. If the players skipped step 1 above, pick one player at random to choose the pattern. (Note: once they have experience with the game, players may invent their own patterns and use those instead. Many thousands of patterns are possible) 3. Setting up an initial position. After choosing the goal pattern the pattern-chooser can set up some initial position by placing any number of stones of any color on the board (or none at all). The purpose of this step is to set up a position that seems to be balanced both for the builder and the blocker. Important remark! You must not form the goal pattern at this step since the player roles aren't chosen yet! If you form the goal pattern at this step then the second player can choose to be builder and win the game. If you are not an experienced player it is better to skip this step when you are the pattern chooser. 4. Choosing a builder and a blocker At this step the player who did not choose the pattern can analyze the chosen pattern and the initial position and decide whether he wants to play as the builder or blocker. Play If the players are playing with a game clock, the builder s clock is started now. Starting with the builder, the players alternate turns, placing a single stone of any color to any empty cell of the board. Usually patterns require the builder to place red stones and the blocker to place blue stones. But this is not true for all the patterns and sometimes it will be advantageous to one or both players to play with all available colors. End of Game The game ends either when the board is completely full or the pattern has been built (but never before step 3 is complete). If the pattern has been built, the builder wins. Otherwise, the blocker wins. The game may also end when a player resigns or runs out of time.

3 Patterns To get you started, we provide a list of patterns below. A few preliminary notes: In each example, the board on which the pattern should be played appears in a pictured illustration. Unless otherwise specified, assume that only two stone colors (red and blue) are available to the players. In order to keep the pattern descriptions succinct, we rely on a few terms which are defined in the glossary below. You needn t read the glossary unless the pattern descriptions confuse you. If you decide to invent your own patterns, there are a couple of rules: the pattern must describe constraints only on the distribution of stones/empty spaces on the board, and nothing else (like, for example, which player placed which stone, or the order in which stones were placed, or the time of day, etc) Glossary Connected Group A set of stones on the board, for which it is possible to trace a continuous path between any two of them by stepping between adjacent stones in the set. Size to the number of stones or empty spaces in a group or pattern of stones or empty spaces. Embedded A connected group of stones is said to be embedded when it is connected to other stones, of the same color, that are not themselves part of the group. Most games wherein players try to build certain kinds of groups implicitly allow the groups to be embedded. For example, in the game Hex, the winning chain of stones connecting opposite sides may be connected to other pieces of the same color that branch off of the main chain, but which aren t themselves involved in the connection. It is important to discuss this property explicitly here, because it can cause confusion when players invent pattern goals. An example will help to illustrate the idea. Let us say that the pattern that the builder must build is a smiley face, such as the one above to the right. Now, let us imagine that during the course of the game, the board ends up in the state illustrated by the next picture. Clearly, although the pattern of red stones is not itself a smiley face, it does contain a smiley face, as can be seen by highlighting some of the stones, as in the picture at right. The question is: has the builder won? Well, if the pattern may be embedded, then yes, the builder has won. He has built a smiley face that is embedded in a larger pattern of red stones. Practically speaking, most balanced patterns may be embedded (one exception are patterns that place constraints on empty spaces). Therefore, if you are unsure whether the builder s pattern can be embedded, assume that it can.

4 Pattern 1: Yin Yang A connected group of stones, consisting of two subgroups. Each of the two subgroups should consist entirely of stones of one color, and that color should be different from that of the other subgroup. The two subgroups should have the same size and shape. The two subgroups should be connected in such a way that the overall group that they form can be rotated 180 degrees so that the two subgroups swap positions with one another. When you propose this pattern, you should specify the minimum size that the builder must achieve. For example, when you propose Yin Yang 12, it means the builder must make a Simple Yin Yang consisting of at least 12 stones. The group may be embedded. Below are two examples that satisfy the requirements for Yin Yang 12 Pattern 2: Almost Y A chain of red stones connecting at least three sides of the board, where at least one of those sides is not adjacent to the other two. Stones in corners are not considered to be connected to any side. The chain may be embedded. An example appears in the picture at right. Pattern 3: Rusty Chain A chain of stones, consisting of either all red stones or all blue, where every triplet of stones in the chain forms a 120 degree angle. Links from different sections of the chain may touch, but may not cross over one another. You should specify a minimum chain length when proposing this pattern, as a way to balance the goals of builder and blocker. Append the minimum chain length to the end of this pattern s name when proposing it. For example, you might say Rusty Chain 12, which means that the builder must build a rusty chain of at least 12 stones. The pattern may be embedded. An example appears at right.

5 Pattern 4: Odd The board must be completely full, and the sum of the number of distinct red connected groups and the number of distinct blue connected groups all larger than size X must be odd. The pattern-chooser also chooses a value of X when choosing this pattern. We recommend choosing X to be 7-9 for the board pictured. An example that satisfies the criteria for Odd 7 appears in the picture (there are five groups consisting of 7 or more stones). Although the board is full in the example, in actual play it will become obvious who the winner will be well before the board is full, so most games will end with a resignation. Pattern 5: Havannah Either a fork, a bridge or a loop of red stones and, at the same time, none of those structures in blue. The structures may be embedded. A fork is a chain linking any three sides. Corners do not belong to sides. A loop is a chain around at least one cell. The cell(s) inside the loop can be empty or occupied by any color. A bridge is a chain linking any two corners. A Bridge A Fork A Loop This is almost identical to the game of the same name, invented by Christian Freeling.

6 Pattern 6: Rigid Shape This is not just one pattern, but a class of patterns. A rigid shape is a group of stones of a specified size (and sometimes color(s)), where the positions of all the stones relative to one another are fixed. An example is a straight line of 6 red stones, as in the first picture at right. Typically (but not necessarily), there are no restrictions on the allowable orientation or position of the rigid shape, and the pattern may be embedded. Another example is an equilateral triangle consisting of 10 red stones, as in the second picture. Figure 1 It is also possible to define rigid shapes in which not all the stones are part of a single connected group. For example, see the smiley face in the third picture. You can propose a rigid shape simply by tracing an example on the board. Rigid shapes are easy to understand, but they are difficult to build unless they are small. Here are 3 ways (among many) to modify the definition of a rigid shape to make it easier for the builder: 1. You can specify that the rigid shape may be built in any available color, so long as all the stones in it are the same color. 2. You can allow there to be a certain number of "holes" in the rigid shape. A rigid shape that has a hole is "missing" a stone that it would need in order to be complete. There is typically no restriction on where the hole must be within the shape, or what must be in the hole- it may be empty or contain a stone of another color. Note that you can introduce holes to ANY pattern, not just rigid shapes, in order to make the builder's job easier. As a result, holes are useful in defining and balancing patterns. 3. You can specify that the builder's pattern can be any one of several rigid shapes. For example, you could specify that the builder's pattern must contain either a 10-stone equilateral triangle, as in figure 2, OR a smiley face, as in figure 3. The more options the builder has, the easier his job is. Here is an example of a pattern proposal for a rigid shape, with modifiers: "An equilateral triangle consisting of 10 stones, either all blue or all red. It may contain up to 2 holes, and it may be embedded." The fourth picture shows an example of the completed pattern. A green border appears around the completed shape so the reader can easily see it amongst the stones in which it is embedded. There are two holes in the shape, one blue, the other empty.

7 Pattern 7: Voltron This is not just one pattern, but a class of them, related to the Rigid Shape class of patterns. A Voltron is a Rigid Shape that may be built in fragments on different parts of the board. To specify a Voltron pattern, first you describe a rigid shape, and then you specify the maximum number of fragments it may be in when built. For example, consider a square loop with four red stones on each side, as illustrated in the first picture. To turn this rigid shape into a Voltron, you can specify that it may be built in up to two fragments. The second and third pictures show two examples that would satisfy this pattern. When proposing a Voltron, it s important to specify whether the orientations of the fragments must all be the same, or whether differing orientations are allowed. To illustrate: the two fragments in the second picture have the same orientation: you wouldn t have to rotate either one if you wanted to bring the two fragments together to form the shape in the first picture. On the other hand, the fourth picture illustrates a case where the same two fragments do not have the same orientation: you would have to rotate one of them if you wanted to form the original shape. You must specify whether this situation counts as a completed pattern. Typically, as with most patterns, Voltrons may be embedded.

8 Pattern 8: Mirror, Mirror There must be two connected groups of red stones on the board, that are not exactly the same shape as one another other, but which are instead mirror images of one another. This means that if you could flip one of the groups over, it would be identical to the other. The two groups may have different orientations, and both may be embedded. The player who proposes the pattern should specify a minimum size for each group. For example, to propose the pattern, he might say Mirror, Mirror 10, which means that the groups must each have at least 10 stones in them. Figure 1 illustrates an example of Mirror, Mirror 10. Pattern 9: XOR Moku The board must be completely full of stones, and must contain either five red stones in a straight line, or five blue stones in a straight line, but not both. Lines of stones may be horizontal, vertical, or diagonal, and they may be embedded. The first two pictures below illustrate examples of 5 stones in a row (but not completed games, because the board is not full), and the third picture shows an example of a completed game where the blocker has won, because both a red and a blue line of 5 stones are present (and this cannot be changed by filling up the rest of the board).

9 Pattern 10: Linear Chaos The board must be completely full, and it must contain more red lines than blue lines. A line is defined here as a straight row of at least X stones (the Pattern-chooser picks the value of X). Note that a line can be oriented in any of three directions on the board (labeled A, B, and C in the picture below). The number of lines on the board are calculated by counting lines in each of the three directions independently. This means that a single stone can be a part of three separate lines (one in each direction). Also, lines which are longer than the minimum length X count only as a single long line, not as multiple shorter lines running in the same direction. For example, let s say that in a particular game, X=4. Let s say also that there is a line of 8 red stones in a row on the board. This counts as only one 8- stone line, not as two 4-stone lines. An example of a finished game and score calculation is given below: B C A Let s say that in this game, X=4. The table below gives the number of red and blue lines in each direction: Direction Red Blue A 6 4 B 5 2 C 4 2 Totals 15 8 In this case, there are more red lines (15) than blue ones (8), so the builder wins.

10 Pattern 11: Reflectology Two red groups of the same size (The player who proposes the pattern must specify the minimum size), which are mirror-symmetric about any one of the three axis lines depicted in the first picture. These groups may be embedded. An example of a Reflectology 7 pattern is shown in the second picture, along with its axis of symmetry. Also note: any stone through which an axis line passes (see the bottom picture for an example) cannot be part of any pair of groups that are symmetrical about that axis, because there are no other spaces with which it could be symmetric.

11 Pattern 12: Triple Hex The Mind Ninja board can be divided into three overlapping sections, as illustrated in the picture below. Notice that each of the sections has two red edges and two blue edges. The builder must build a chain of red between the two red edges on at least two of the sections, for a total of at least two red chains. Each chain must stay entirely within the section whose sides it connects. The chains may be embedded. An example of the completed pattern is given below. If you are familiar with the game Hex, this is like playing three simultaneous games of Hex on three partially overlapping boards.

12 Pattern 13: Bre-e-edge The board must be completely full, and when it is, the red stones must have more points than the blue stones. Points are assigned as follows: A color receives X points for every one of its stones occupying an edge or corner space, and -Y points for every separate group of that color on the board. The patternchooser sets the values for X and Y. A group may be any size, as small as a single stone, and it may contain stones occupying edge points. The tallies for edge points and group points are independent, so edge stones will be counted twice. An example is given using the figure below. Let s say that the pattern-chooser set X=1, and Y=-3. Red has (17 edges stones)*1 = 17 Red has (3 groups)*-3 = = 8 Blue has (19 edges stones)*1 = 19 Blue has (10 groups)*-3 = = -11 So the red stones have 8 points, the blue stones have -11, and the builder wins. Pattern 14: Product War The board must be full, and the product of the sizes of the two largest red groups must be larger than the product of the sizes of the two largest blue groups. If there is only one group in a given color, the product is defined as zero. The picture at right shows an example in which the product of the two largest red groups is 27*46=1242 and the product of the two largest blue groups is 7*30=210. Therefore, the builder has won.

13 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. 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 of the patterns must be formed. Thus, neither draws nor infinite positional repetitions can occur. 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.

14 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 free turns in exchange 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 Havannah, Y, Star, 5-In-a-Row, and One-Capture-Go are all contained as specific patterns in Mind Ninja. In fact, Mind Ninja contains nearly all games where players fill in spaces on a board in order to form some kind of pattern.