# Number Scrabble

Last updated

Number Scrabble (also known as Pick15 [1] [2] [3] or 3 to 15 [4] ) 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 to amass a personal total of exactly 15 wins the game. [5] [6] The game is isomorphic to tic-tac-toe, as can be seen if the game is mapped onto a magic square. [6]

## Play

Number Scrabble is played with the list of numbers between 1 and 9. Each player takes turns picking a number from the list. Once a number has been picked, it cannot be picked again. If a player has picked three numbers that add up to 15, that player wins the game. [5] [6] [7] However, if all the numbers are used and no player gets exactly 15, the game is a draw. [5] [6]

The game is identical to tic-tac-toe, as can be seen by reference to a 3x3 magic square: if a player has selected three numbers which can be found in a line on a magic square, they will add up to 15. If they have selected any other three numbers, they will not. [8]

## Related Research Articles

Tic-tac-toe, noughts and crosses, or Xs and Os, is a paper-and-pencil game for two players, X and O, who take turns marking the spaces in a 3×3 grid. 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.

Combinatorial game theory (CGT) is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field.

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×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×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 4x4x4. 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".

In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game in which an extra move can never be a disadvantage.

Time Machine is an American game show where contestants compete to answer trivia questions about popular culture and recent history to win prizes. The show aired on NBC from January 7 through April 26, 1985, and was hosted by John Davidson. Charlie Tuna was the announcer, with Rich Jeffries as his substitute. Reg Grundy Productions produced the series, and upon its premiere Time Machine was one of three Grundy series airing on NBC.

SOS is paper and pencil game for two or more players. It is similar to tic-tac-toe and dots and boxes, but has greater complexity.

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.

Quantum tic-tac-toe is a "quantum generalization" of tic-tac-toe in which the players' moves are "superpositions" of plays in the classical game. The game was invented by Allan Goff of Novatia Labs, who describes it as "a way of introducing quantum physics without mathematics", and offering "a conceptual foundation for understanding the meaning of quantum mechanics".

Harary's generalized tic-tac-toe or animal tic-tac-toe is a generalization of the game tic-tac-toe, defining the game as a race to complete a particular polyomino on a square grid of varying size, rather than being limited to "in a row" constructions. It was devised by Frank Harary in March 1977, and is a broader definition than that of an m,n,k-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.

Little League World Series Baseball is a series of sports video games. Based on the Little League World Series, there are three games in the series. No game in the series was released after 2010.

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 in the smaller tic-tac-toe boards until one of them wins in the larger tic-tac-toe 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.

Wild tic-tac-toe is a 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.

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. Harary's generalized tic-tac-toe is an even broader generalization. The game can also be generalized as a nd game. 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.

A nd game (or nk game) is a generalization of the 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.

Combinatorial Games: Tic-Tac-Toe Theory is a monograph on the mathematics of tic-tac-toe and other positional games, written by József Beck. It was published in 2008 by the Cambridge University Press as volume 114 of their Encyclopedia of Mathematics and its Applications book series (ISBN 978-0-521-46100-9).

The Matchbox Educable Noughts and Crosses Engine or MENACE was an analogue computer made from 304 matchboxes designed and built by Donald Michie in 1961. It was designed to play human opponents in games of noughts and crosses by returning a move for any given state of play and to refine its strategy through reinforcement learning.

## References

1. Michon, John A. (1 January 1967). "The Game of JAM: An Isomorph of Tic-Tac-Toe". The American Journal of Psychology. 80 (1): 137–140. doi:10.2307/1420555. JSTOR   1420555. PMID   6036351.
2. Simon, Herbert A. (1969). The Sciences of the Artificial. MIT Press. ISBN   9780262264495. Here are the rules of a game, which I shall call number scrabble.
3. Cazenave, Tristan; Winands, Mark H. M.; Edelkamp, Stefan; Schiffel, Stephan; Thielscher, Michael; Togelius, Julian (2016-05-11). Computer Games: Fourth Workshop on Computer Games, CGW 2015, and the Fourth Workshop on General Intelligence in Game-Playing Agents, GIGA 2015, Held in Conjunction with the 24th International Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, July 26-27, 2015, Revised Selected Papers. Springer. ISBN   9783319394022.
4. Ham, Ethan (2015-06-19). Tabletop Game Design for Video Game Designers. CRC Press. ISBN   9781317536048.
5. Juul, Jesper (2011-08-19). Half-Real: Video Games Between Real Rules and Fictional Worlds. MIT Press. ISBN   9780262516518.
6. "TicTacToe Magic" (PDF). December 11, 2016. Retrieved December 11, 2016.
7. "Fifteen : nrich.maths.org". nrich.maths.org. Retrieved 2016-12-11.
8. Math!, Oh Boy I. Get To Do (2015-05-30). "Oh Boy! I Get to do Math!: Tic-Tac-Toe as a Magic Square". Oh Boy! I Get to do Math!. Retrieved 2016-12-11.