# THE MOKU FAMILY

(with collaboration of Edward Jackman and Bill Taylor)

N in-a-row games (win by making a line of N connected stones) seem to appear independently in different cultures, which is not very common to abstract games.

ROWS OF 3

The most basic game of this family, played by most children, is Tic-Tac-Toe played in a 3x3 board:

 DROP - Each player, at each turn, drops a stone on an empty cell. GOAL - The winner is whoever makes a 3 in-a-row (orthogonally or diagonally).

It's easy to see that this game always end in a draw, with reasonable moves. Some complex variants: 3 Musketeers, Dara, Wali, Neutreeko, 3 Stones, Winkers, 4D TTT and Shift TTT
(where each row may be shifted left or right, so that the stones in the upper rows may fall down).

Another variant from Richard Hutnik (c) 1995, is Dual Tic-Tac-Toe: Select a board with size NxM (N>2 and M>3). Each player drops a stone of either color on an empty cell. The first player that makes a (orthogonal or diagonal) 3 in-a-row of either color wins. If the board gets full with no 3 in-a-rows, the game is a draw. Some variants are possible: Misère (who makes a 3 in-a-row loses), use more than two colors, more than two players...

A 1985 variant called King by Jim Wilkinson, uses six pieces (three of each color) with a setup phase where both players alternatively drop their stones into empty cells (the first drop cannot be at the center). After all pieces are on board, players move one piece per turn into a (diagonal or orthogonal) adjacent empty cell until one of them make a friendly 3 in-a-row. There is a move exception in the rules: "except diagonally black-to-black" which may only make sense with a view of the board (which I do not have).

There are several 3 in-a-rows in 3D (also check Qubic in the next section). Two of those are 1972's Rex and 1978's Trio. In Rex, there are three set of three columns where it is possible to drop up to 3 pieces (of either color) on each. Wins the player that achieves the first 3 in-a-row (either vertically, horizontally or diagonally in 3D). In Trio, the board is a 3x2x3 column pattern:

o o o
o o
o o o

both players try to achieve as many 3 in-a-rows possible, winning the player with the largest number.

Shift TicTacToe is a 1988 variant where: Two players play Tic-Tac-Toe on a "Connect Four" type board (meaning placing a disk sends it to the bottom first). On their turn, a player can drop a disc in, or slide a row one space to the left or right. Since the board is a 3x3, slides can sometimes send pieces out of the unit and back into regular play. The first to achieve 3-in-a-row is the winner.

Flip Tic-Tac-Toe (a 2006 game by Stratagem Publishers) is played on a 4x4 board with 64 stackable stones (4 colors with 16 stones each which I assume allows 2-4 players). Players are able to drop friendly stones (on empty cells or increasing stacks already on board), can move stacks (orthogonally to empty cells?) and also flip an entire stack (the bottom becomes to top). The goal is to make a 3 in-a-row of your own color.

ROWS OF 4

There are several games that use 4 in-a-row, the most famous is Connect 4 (also check Contigo, Delta, Firedrill, Gravity, Magneton, Complica). One variant, called QUBIC (Parker Brothers, 1965) is played in 3D on four 4x4 boards, winning the first to make a 4 in-a-row (this game has other names, like Tramp).

 DROP - Each player, at each turn, drops a stone on an empty cell. GOAL - The winner is whoever makes a 4 in-a-row (orthogonally or diagonally in any direction).

Both Patashnik (around 1980) and independently Louis Victor Allis and Patrick Shoo (around 1991) solved Qubic (first player win). Allis also solved Connect-4 (check this paper). To play some Qubic games check this webpage.

Quintego is a Connect-4 variant (with 5 columns): when a vertical column has six stones, and another stone is added, the bottom stone is removed and returned to the player's reserve (and all the others on that column move down). I think another name for this game is Complica.

Another interesting variant is Gigamic's Quarto. In this game, there are 16 different pieces (all combinations of 4 boolean features: high/low, round/square, light/dark and solid/hollow). A player drops a piece on an empty cell and selects the next piece which must be dropped by the opponent! The first player to make a 4 in-a-row of any of the 4 features, wins.

This game can be played with the following 16 stacks, where each feature is given by the color of a stone at a certain height.

A nice 3 player variant can be played (on a Go board) with the addition of the following rule: each player is obliged to block a winning threat of the next player.

 An 3 player example With the next-block rule it's easier to be aggressive, knowing the next player's move may be partly restricted. The player just needs to force a win (e.g. a simultaneous double-open 3 and another 3).  It is very dangerous to make a single-ended 3, because if the next player has a double-ended 2, he will extend it to 3, leaving the 3rd player compelled to block the 1st, so then there is only one player left to block the double 3. In the left diagram (white-black-red turns), Black should not play at [1] since Red drops at [2], then White is obliged to block the black 3 in-a-row and Black can only block one side of the Red's already winning line.

ROWS OF 5

Perhaps the most famous 5 in-a-row game is GO-MOKU, an ancient Japanese game. The full name, "Go-moku Narabe," literally means "five stones in a row." In Japan, a more complex version of the game exists, known as Renju. Check more information at playSite. Victor Allis' Thesis presents the solution of Go-moku (first player wins).

The game starts at an empty Go board.

 DROP - Each player, at each turn, drops a stone on an empty cell. GOAL - Make a line of 5 connected stones (horizontally, vertically or diagonally).  If a player makes a line of 6 or more stones, he does not win and the game continues!

 An example Black's turn. White has a winning position at the three cells [1].

Murray in his book "A history of board-games other than chess" refers another set of rules (game 3.7.1) consisting of two phases. First a drop phase where both players drop an agreed number of stones, then a move phase where each player moves a friendly stone into an adjacent (orthogonal) empty cell. Win by making an orthogonal 5 in-a-row. A similar variant (using twelve stones) states that each time a (orthogonal or diagonal) 5 in-a-row is achieved (in the drop or move phase), the player removes an enemy stone. The player which reduces the adversary to four stones, wins the game.

A traditional variant called Gomoku-Ninuki adds an extra custodian capturing rule: two enemy stones are captured when placed between two friendly stones, and a new winning condition: a player also wins by capturing 10 stones. This game is also known as Pente. Bolix is a similar variant played on a 5,10,5 hexagonal board, but where only one of the two enemy stones is captured.

In the end of the XIXth century, Japanese players started to change Go-moku and tried to create a more balanced game (the first player, Black, has a tremendous advantage) which they called RENJU (read the game's history). This was done by creating some restrictions in Black's move, namely:

• Black cannot create a 2 open 3 lines (and so winning)
• Black cannot create a 2 closed 4 lines (and so winning again)
• Black cannot create an overline (i.e., 6 or more stone line)
• Black's 1st move is a the center board, and Black's 2nd move must be at cell distance greater than 2.
• White's 1st move is decided by Black.
• At White 2nd move, there is the PIE rule: White may decide to change sides!

For more detailed information, check the international rules of Renju maintained by RIF: The Renju International Federation (some variants are also shown).

There are a lot of variations of this simple game. A curious crossover between Gomoku and Reversi, called Gomullo invented by Roland Johansson at 1999 and can be defined as: wins who gets N stones in-a-row in a NxN-board, using the rules of Reversi. Another variant, with shifting rows, is Perplexus, where the player may drop a stone or shift a row one cell to the right or left (with 6 sliding rows of 7 stones placed within a base row of length 11). Other games: Andantino, Kassle, Quixo, Campaign, Ergo, Pentagonal 3D5 and Hex-gomoku.

Some comments from Edward Jackman: Here's a list of modifications we've used to address first player advantage and handicapping:

 Pie rule -- First player moves, second player picks which side to play. Three move equalization -- after three turns, second player may swap sides. Neutral stones -- White gets one neutral stone. At any turn, she may play both a normal stone and her neutral stone. The neutral blocks both players. (Another game -- give both players several neutrals -- say 4 for black, 6 for white. Play normal stone plus ANY NUMBER of neutrals per turn. Four-in-a-row wins.) Blacks first stone is a 'half stone'. It may only be part of a winning *diagonal* row. If black forms a non-diagonal 5-in-a-row using that stone, the game continues. Double move. Black plays one stone; thereafter each player plays two stones. Restricted double move. Black plays one stone; thereafter each player plays two stones that are not on the same row, column or diagonal within 5 spaces of each other.  Add capturing rules as in Ninuki-renju or Keryo-Pente, etc. White can be required to capture fewer stones than black for the win. Alternately, captures have no effect on the outcome -- 5-in-a-row is the only way to win.  Black has fewer stones. Each player has a limited number of pieces, maybe only 15 or 20. Black has fewer pieces than white. Each turn a player may place a stone or move a stone from any cell to any other. Optionally, play with unlimited stones, but every Nth move, black moves a stone rather than placing another.

Another game from Edward (around 1995) is Progressive Gomoku: Black starts by dropping one stone, then White drops two, then Black drops three... There is, however, one restriction: during a drop sequence, all the stones must be mutually non-adjacent (i.e., they cannot be adjacent to each other).

Next, two very playable variants (played in large enough boards) designed by Bill Taylor.

First, a definition:

 DAGGER - A player with a dagger can, in his turn, drop two stones on two empty cells. The dagger can not be used to make a 5 in-a-row. The dagger cannot be used twice consecutively by same player, except to defend against an immediate loss. When the dagger is used, it goes to the other player..

DAGGER GOMOKU - Same as Go-moku except that the second player starts with a dagger:

A second game is called QUAD-MOTU:

 PIE RULE - One player drops two black stones and one white stone. The other player choose sides. The next drop is a white stone. TURN - In each turn, each player may: Drop a stone on an empty cell. (optionally) Move a stone to an (orthogonal or diagonal) adjacent empty cell. GOAL - The winner is whoever makes a 5 in-a-row.

This rules define a more "violent" game. This extra freedom may provide a solution to the drawish feature of gomoku in hex boards.

Another related variant is to give each player just 12 stones. After the drop phase, the move phase begins: each player moves one stone to an adjacent (orthogonal or diagonal) empty cell. Wins the player that makes a 5 in-a-row.

Applying the progressive mutator (drop one stone at the first move, then drop two, then drop three...)  to Gomoku also provides a very enjoyable game, if we add a restriction: No two stones in a turn may finish in the same group (i.e., a set of friendly stones orthogonally or diagonally connected). If a fast game, but with lots of tactical subtleties. This game should be played on an unlimited board.

 An example White's turn. This is the 8th move, so White can drop eight stones. However, it is enough to drop four (the marked stones).  After that sequence, Black cannot stop the double attack at cells [1] because it would violate the restriction (two stones on the same group).

Tomas Flodén designed Pentago, a 5 in-a-row game where a 6x6 board is made out four separate 3x3 boards. Each player, on his turn, drops a friendly stone on an empty cell and then rotates one of the four small board 90 degrees in any direction (i.e., clockwise or counterclockwise). If all cells are occupied before a 5 in-a-row is achieved, the game is a draw. Also, if due to a rotation, both player get a 5 in-a-row, the game is also a draw. This idea suggests a hexagonal variant, where each cell on the board could be select as a pivot to rotate the adjacent 6 cells in either direction (a physical board would be difficult to make, it would be better to play this variant on a computer).

A 2005 game which also tries to balance first player advantage without protocol moves is Vincent Everaert's Love Gomoku . On each turn, each player drops a friendly stone on an empty cell and then an enemy stone on an adjacent empty cell. If there is no additional empty cell to drop the enemy stone, the turn is invalid. As usual, a friendly 5 in-a-row wins the game.

 An example [puzzle by the author] Black wins in two moves. How? Answer: 1.h6,g5 forcing White to reply g6,h5 and winning with h8 (select this line) .

ROWS OF 6

Games with more than 5 in-a-row need extra liberties to avoid easy draws (it's much easier to block a player in a regular board). One possibility is to add the 1222 progressive mutator: In each turn, each player drops two stones (except in the first turn, which balances the game).

 An example The marked black stone has coordinates (0,0). All the other drops are relative to that first one (this notation is used because the board is not limited). 1.  0,0           1,0   1,-1 2.  1,2   0,2    -1,-1 -1,3 3.  0,1   0,-1    0,-2  0,3 4. -2,1  -1,1    -3,1   3,1 5. -2,0   1,3    -3,-1  2,4 6. -1,0  -3,2     1,-2 -4,3 7. -2,-1  1,2    -3,-2  2,3 8. -2,2  -1,2      1-0

This game has the advantage of nullifying the opening advantage by 1222 play. Bill Taylor thinks it is a forced win, but has no idea to whom! In practice games tend to be a similar length to normal Go-moku.

There is a 1983 game, Jürgen Hagedorn's Hexago to make a 6 in-a-row on a large hexagonal board, however, with no additional rule, this simple goal cannot be done.