# Tic-tac-toe variants

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Tic-tac-toe is an instance of an m,n,k-game, where two players alternate taking turns on an m×n board until one of them gets k in a row. [1] Harary's generalized tic-tac-toe is an even broader generalization. The game can also be generalized as a nd game. [2] The game can be generalised even further from the above variants by playing on an arbitrary hypergraph where rows are hyperedges and cells are vertices.

## Contents

Many board games share the element of trying to be the first to get n-in-a-row, including three men's morris, nine men's morris, pente, gomoku, Qubic, Connect Four, Quarto, Gobblet, Order and Chaos, Toss Across, and Mojo.

Variants of tic-tac-toe date back several millennia. [3]

## Historic

An early variation of tic-tac-toe was played in the Roman Empire, around the first century BC. [4] It was called Terni Lapilli and instead of having any number of pieces, each player only had three; thus, they had to move them around to empty spaces to keep playing. The game's grid markings have been found chalked all over Rome. [5] However, according to Claudia Zaslavsky's book Tic Tac Toe: And Other Three-In-A Row Games from Ancient Egypt to the Modern Computer, Tic-tac-toe could be traced back to ancient Egypt. [6] [7] Another closely related ancient game is three men's morris, which is also played on a simple grid and requires three pieces in a row to finish. [8]

## Variants in higher dimensions

### 3D Tic-tac-toe

Three-dimensional tic-tac-toe on a 3×3×3 board. In this game, the first player has an easy win by playing in the centre if two people are playing.

One can play on a board of 4x4 squares, winning in several ways. Winning can include: four in a straight line, four in a diagonal line, four in a diamond, or four to make a square. Another variant, Qubic, is played on a 4×4×4 board; it was solved by Oren Patashnik in 1980 (the first player can force a win). [9] Higher-dimensional variations are also possible. [10]

The 3D tic tac toe variant does not always appear as 3 dimensional board. Some variants have different forms. For example in the case of Gobblet-like variant, Gobblet Gobblers [11] and Otrio, [12] the third element appears as variation in marker sizes (small, medium, large). Players can 'steal' the opponent spot by placing larger marker at the top of the opponent smaller marker or just simply competing with overlapping spot.

## Misère games

### Misere Tic-tac-toe

In misère tic-tac-toe, the player wins if the opponent gets n in a row. [13] [14] [15] [16] This game is also known as avoidance tic tac toe, [14] toe-tac-tic, [14] [17] inverse tic tac toe, [15] or reverse tic tac toe. [16] A 3×3 game is a draw. More generally, the first player can draw or win on any board (of any dimension) whose side length is odd, by playing first in the central cell and then mirroring the opponent's moves. [10] [15]

### Notakto

Notakto is a misere and impartial form of tic tac toe. This means unlike in misere tic tac toe, in Notakto, both players play as the same symbol, X. [18] It also can be played on one or multiple boards. [19]

## Variants with bigger boards

### Quixo

The game Quixo is played on a five-by-five board of cubes with two players or teams. [20] On a player's turn, they select a blank cube or a cube with their symbol on it that is at the edge of the board. If a blank cube was selected, the cube is turned to be the player's symbol (either an X or O). The game ends when one player gets five in a row. [20] [21] [22] [23]

### Unrestricted n-in-a-row

Unrestricted n-in-a-row is played on an infinite tic-tac-toe board where the goal is for one player to get n in a row. [2]

### Amőba

The game called Amőba (amoeba) in Hungary is played on squared paper; it is a five-in-a-row variant. The winner of a match gets to fence in the completed game with a tight continuous line resulting in an amoeba-looking shape, hence the name. [24]

### Ultimate tic-tac-toe

In Ultimate tic-tac-toe, the board is composed of a large tic-tac-toe board where each cell contains another standard tic-tac-toe board. A move in the smaller boards determines the location of the next move in the larger board. [25]

## Isomorphic games

### Number Scrabble

There is a game that is isomorphic to tic-tac-toe, but on the surface appears completely different. It is called Pick15 [26] or Number Scrabble. [27] Two players in turn say a number between one and nine. A particular number may not be repeated. The game is won by the player who has said three numbers whose sum is 15. [26] [28] If all the numbers are used and no one gets three numbers that add up to 15 then the game is a draw. [26] Plotting these numbers on a 3×3 magic square shows that the game exactly corresponds with tic-tac-toe, since three numbers will be arranged in a straight line if and only if they total 15. [29]

### Word Tic-tac-toe

Another isomorphic game uses a list of nine carefully chosen words, for instance "eat", "bee", "less", "air", "bits", "lip", "soda", "book", and "lot". Each player picks one word in turn and to win, a player must select three words with the same letter. The words may be plotted on a tic-tac-toe grid in such a way that a three in a row line wins. [30]

## Dexterity Variants

Tic tac toe can be played by integrating element of dexterity to place the markers. Objects such as balls can be thrown to a grid (which can be made from other objects such as glasses) to get three marks in a row, leaving elements of probability for the markers to be landed at the intended spot and stimulating physical exercises. [31] [32] [33]

## Other Variants

### Numerical Tic-Tac-Toe

Numerical Tic Tac Toe is a variation invented by the mathematician Ronald Graham. [34] The numbers 1 to 9 are used in this game. The first player plays with the odd numbers, the second player plays with the even numbers. All numbers can be used only once. The player who puts down 15 points in a line wins (sum of 3 numbers). [35] This game can be generalized to a n × n board. [35]

### Check Lines

In the 1970s, there was a two-player game made by Tri-ang Toys & Games called Check Lines, in which the board consisted of eleven holes arranged in a geometrical pattern of twelve straight lines each containing three of the holes. Each player had exactly five tokens and played in turn placing one token in any of the holes. The winner was the first player whose tokens were arranged in two lines of three (which by definition were intersecting lines). If neither player had won by the tenth turn, subsequent turns consisted of moving one of one's own tokens to the remaining empty hole, with the constraint that this move could only be from an adjacent hole. [36]

### Twice crosses-circles

• Programmed in 1989, the algorithm was previously tested on the Elektronika MK-52. There is also a variant of the game with the classic 3×3 field, in which it is necessary to make two rows to win, while the opposing algorithm only needs one. [37]

### Quantum Tic-Tac-Toe

Quantum tic tac toe allows players to place a quantum superposition of numbers on the board, i.e. the players' moves are "superpositions" of plays in the original classical game. This variation was invented by Allan Goff of Novatia Labs. [38]

### Wild Tic-Tac-Toe

In wild tic-tac-toe, players can choose to place either an X or O on each move. [7] [39] [40] [41] It can be played as a normal game where the player who makes three in a row wins or a misere game where they would lose. [7] This game is also called your-choice tic-tac-toe [42] or Devil's tic-tac-toe.[ citation needed ]

### SOS

In the game SOS, the players on each turn choose to play a "S" or an "O" in an empty square. [43] If a player makes the sequence SOS vertically, horizontally or diagonally they get a point and also take another turn. [44] The player with the most points (SOSs) is the winner. [43] [44]

### Treblecross

In Treblecross, both players play with the same symbol (an X [15] or black chip [45] ). The game is played on a 1-by-n board with k equal to 3. [15] The player who makes a three in a row of Xs (or black chips) wins the game. [15] [45]

### Revenge n-in-a-row

In revenge n-in-a-row, the player who makes an n-in-a-row wins unless the opponent can make an n-in-a-row in the next move where they lose. [46] [15]

### Random turn Tic-Tac-Toe

In the game random turn tic-tac-toe, a coin flip determines whose turn it is. [7]

### Quick Tic-Tac-Toe

In quick-tac-toe,[ clarification needed ] on each turn the players can play their mark in any squares they want provided that all the marks are in the same vertical or horizontal row. The winner is the player who places the last mark. [47]

## Related Research Articles

Tic-tac-toe, noughts and crosses, or Xs and Os is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with X or O. The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row is the winner. It is a solved game, with a forced draw assuming best play from both players.

TacTix is a two-player strategy game invented by Piet Hein, a poet well known for dabbling in math and science, best known for his game Hex.

Three men's morris is an abstract strategy game played on a three by three board that is similar to tic-tac-toe. It is also related to six men's morris and nine men's morris. A player wins by forming a mill, that is, three of their own pieces in a row.

A solved game is a game whose outcome can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory and/or computer assistance.

Combinatorial game theory measures game complexity in several ways:

1. State-space complexity,
2. Game tree size,
3. Decision complexity,
4. Game-tree complexity,
5. Computational complexity.

An m,n,k-game is an abstract board game in which two players take turns in placing a stone of their color on an m-by-n board, the winner being the player who first gets k stones of their own color in a row, horizontally, vertically, or diagonally. Thus, tic-tac-toe is the 3,3,3-game and free-style gomoku is the 15,15,5-game. An m,n,k-game is also called a k-in-a-row game on an m-by-n board.

3D tic-tac-toe, also known by the trade name Qubic, is an abstract strategy board game, generally for two players. It is similar in concept to traditional tic-tac-toe but is played in a cubical array of cells, usually 4×4×4. Players take turns placing their markers in blank cells in the array. The first player to achieve four of their own markers in a row wins. The winning row can be horizontal, vertical, or diagonal on a single board as in regular tic-tac-toe, or vertically in a column, or a diagonal line through four boards.

In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".

Toss Across is a game first introduced in 1969 by the now defunct Ideal Toy Company. The game was designed by Marvin Glass and Associates and created by Hank Kramer, Larry Reiner and Walter Moe, and is now distributed by Mattel. It is a game in which participants play tic-tac-toe by lobbing small beanbags at targets in an attempt to change the targets to their desired letter. As in traditional tic-tac-toe, the first player to get three of their letters in a row wins the game. There are other similar games to Toss Across known under different names, such as Tic Tac Throw.

Dara is a two-player abstract strategy board game played in several countries of West Africa. In Nigeria it is played by the Dakarkari people. It is popular in Niger among the Zarma, who call it dili, and it is also played in Burkina Faso. In the Hausa language, the game is called doki which means horse. It is an alignment game related to tic-tac-toe, but far more complex. The game was invented in the 19th century or earlier. The game is also known as derrah and is very similar to Wali and Dama Tuareg.

Achi is a two-player abstract strategy game from Ghana. It is also called tapatan. It is related to tic-tac-toe, but even more related to three men's morris, Nine Holes, Tant Fant, Shisima, and Dara, because pieces are moved on the board to create the 3-in-a-row. Achi is an alignment game.

Picaria is a two-player abstract strategy game from the Zuni Native American Indians or the Pueblo Indians of the American Southwest. It is related to tic-tac-toe, but more related to three men's morris, Nine Holes, Achi, Tant Fant, and Shisima, because pieces can be moved to create the three-in-a-row. Picaria is an alignment game.

Zillions of Games is a commercial general game playing system developed by Jeff Mallett and Mark Lefler in 1998. The game rules are specified with S-expressions, Zillions rule language. It was designed to handle mostly abstract strategy board games or puzzles. After parsing the rules of the game, the system's artificial intelligence can automatically play one or more players. It treats puzzles as solitaire games and its AI can be used to solve them.

Ultimate tic-tac-toe is a board game composed of nine tic-tac-toe boards arranged in a 3 × 3 grid. Players take turns playing on the smaller tic-tac-toe boards until one of them wins on the larger board. Compared to traditional tic-tac-toe, strategy in this game is conceptually more difficult and has proven more challenging for computers.

Notakto is a tic-tac-toe variant, also known as neutral or impartial tic-tac-toe. The game is a combination of the games tic-tac-toe and Nim, played across one or several boards with both of the players playing the same piece. The game ends when all the boards contain a three-in-a-row of Xs, at which point the player to have made the last move loses the game. However, in this game, unlike tic-tac-toe, there will always be a player who wins any game of Notakto.

Number Scrabble is a mathematical game where players take turns to select numbers from 1 to 9 without repeating any numbers previously used, and the first player with a sum of exactly 15 using any three of their number selections wins the game. The game is isomorphic to tic-tac-toe, as can be seen if the game is mapped onto a magic square.

Wild tic-tac-toe is an impartial game similar to tic-tac-toe. However, in this game players can choose to place either X or O on each move. This game can also be played in its misere form where if a player creates a three-in-a-row of marks, that player loses the game.

A nd game (or nk game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions. It is a game played on a nd hypercube with 2 players. If one player creates a line of length n of their symbol (X or O) they win the game. However, if all nd spaces are filled then the game is a draw. Tic-tac-toe is the game where n equals 3 and d equals 2 (3, 2). Qubic is the (4, 3) game. The (n > 0, 0) or (1, 1) games are trivially won by the first player as there is only one space (n0 = 1 and 11 = 1). A game with d = 1 and n > 1 cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line.

Treblecross is a degenerate tic-tac toe variant. The game is an octal game, played on a one-dimensional board and both players play using the same piece. Each player on their turn plays a piece in an unoccupied space. The game is won if a player on their turn makes a line of three pieces in a row.

A strong positional game is a kind of positional game. Like most positional games, it is described by its set of positions and its family of winning-sets. It is played by two players, called First and Second, who alternately take previously untaken positions.

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