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  Dominions Rules 

General

Each player has one six: it constitutes all the ways six beams can radiate from the center of the piece to its sides.

The one six:

Each player has six fives: they constitute all the ways five beams can radiate from the center of the piece to its sides.

The six fives:

Each piece, if flipped around its horizontal axis, displays the same pattern in the opponent's color:

The implication is that the pieces have a fixed orientation, consistent with the 3-D display of the beams. In the applet this is a non-existent problem, but if you make a set at home, unless you actually copy the 3-D display, it is best to put a circle in the middle with an arrow for orientation, as shown here:

Each player has fifteen fours: they constitute all the ways four beams can radiate from the center of the piece to its sides.

There are six compact fours:

There are six forked fours:

Each player has twenty threes: they constitute all the ways three beams can radiate from the center of the piece to its sides.

There are six compact threes:

There are twelve forked threes:

There are two alternating threes:

Each player has fifteen two's: they constitute all the ways two beams can radiate from the center of the piece to its sides.

There are six compact two's:

There are six bent two's:

There are three straight two's:

Each player has six one's: they constitute all the ways one beam can radiate from the center of the piece to its sides.

The six one's:

Each player has one blank: it constitutes all the ways no beams can radiate from the center of the piece to its sides.

The one blank:

The blank plays an important role in the puzzles but not in the game of Dominions.

The Rules

• A group consists of a single piece or several pieces of the same color that are connected by beams. Adjacent blank sides do not constitute a group connection
• The liberties of a group are the beams that border on vacant cells. The pieces of a group share the group's liberties. A group lives if it has at least one liberty
• If a piece, or the group that it is part of, loses its last liberty - its last beam bordering on a vacant cell, the piece or group is reversed, uniting its captors in a new group. The capturing piece may, at the moment of placement, have no liberties itself, but the captured group effectively unites its captors into one group, and this new group must at least have one liberty, see oscillation

Reversal must be done along the horizontal axis. The pieces then reveal the same beam pattern in the opposing color. If you consider actually making a set, it's good to note that the backside pattern is not  the 'see-through' image of the front.

• During the game pieces of both colors get connected by beams. In any position, all beams together (disregarding color) are called the beam structure
• Pieces may be placed without connecting to the existing beam structure. That way disconnected parts of the beam stucture are created. These parts are called sections
• Sections of course may merge by placement of pieces that connect them

• White starts by placing one piece on the board. Players next move in turn to place one piece. Moving is not compulsory: a player may pass his turn without losing the right to move on his next turn. The game ends if both players pass on successive turns
• A piece that is placed must always match neighbours of both colors in terms of beams and blanks
• The piece may not cause oscillation, that is, if it makes a capture, the resulting new group must at least have one liberty!
• The piece must be placed adjacent to at least one opponent's piece - this may be a beam-to-beam or a blank-to-blank contact.
  There is one exception to this condition: A player may freely extend from his own piece or group if this piece or group makes up a complete section of the beam structure! To "extend" means that the piece must add to this section

Moving is not compulsory, but a player who trails in terms of territory cannot afford to pass, since his opponent would pass also and win the game.

• Suicide is legal: if a piece by placement takes all its own liberties - or creates a group with no liberties - it commits suicide and (the group) is reversed immediately. This constitutes the end of the turn. Suicide may not cause oscillation

Suicide has many tactical applications: it may be used to reduce the number of liberties of an opponent's group one aims to attack, while preventing extensions. It may also be used to create places to start a new section. A a piece starting a new section makes up a complete section of the beam structure and thus may be extended freely from by its owner! The nearest example is black commiting suicide on his first move: the resulting white group makes up a complete section and may be freely extended from by white.
In the above example, white has started with the 'six' and black has put a 'five' on top. White next has sacificed a 'one' by suicide to create space to start a new section. Black follows up with his 'six' at the other end, and white starts up a new section with a 'five'. This piece demands black's immediate attention because it makes up a complete section of the beam structure and thus may be extended freely from by white! Again a black suicide is no option: the resulting white group would still make up a complete section. So black is forced to engage in local conflict.

• Oscillation is illegal: placement of a piece may not create a group that is dead in either color

If the first piece on the board has only one liberty, the second player must capture it, but he cannot do so with the matching 'one': the resulting group would be dead in either color and thus 'oscillate'. Oscillation is not restricted to this trivial example: in the next diagram neither   nor   may be placed at 'A'. Both players could place another matching piece (theoretically there are 7 more matches for both), therewith creating a group of five of one color, but if neither would have another matching piece (which can easily happen in the endgame with severely reduced material), 'A' would be an "eye by oscillation", and all pieces would live unconditionally.

The rule that prohibits oscillation of course effectively excludes the blanks from the game.

The score system implies that each player would like to end the game with as little pieces as possible 'in hand'. Pieces that in the end can only commit suicide do not affect the outcome: the moving player gains a point by getting rid of the piece, but so does the opponent because of the extra point of territory.

If a group has a liberty at a cell for which the opponent has no longer a matching piece, this cell is called an 'eye'. A group with an eye lives unconditionally. An eye of one player's group(s) is not necessarily at the same time an eye of the opponent's group(s) that may have liberties at it: one player may still have matching pieces for a cell, while the other doesn't.
Even if both players have a matching piece for a particular cell, it may have 'eye quality': for instance if both players in the above diagram would have only   respectively   to match, both could capture, but the resulting group would only have one liberty and though it would make up a single colored section, from which the captor might freely extend, it would make an easy prey in most circumstances.

Special attention should be given to borderplay: towards the endgame, opportunities for creating eyes and starting new live groups, increase and the demand for pieces with fewer liberties rises with the possibilities to keep them alive. Strength hides in the details!

Compact solutions of the square set are at the core of BackSlide, a "15-puzzle" with a very unusual twist.

The sets quite extraordinary properties also include a one-to-one correspondence with the 64 hexagrams of the I Ching, the Chinese Book of Changes and the possibility to create arrangements called transcendental solutions. In      these properties are used to reveal yet another mystery of a great book.