 m,n,kgame

An m,n,kgame 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.^{[1]}^{[2]} Thus, tictactoe is the 3,3,3game and freestyle gomoku is the 19,19,5game. m,n,kgame is also called as kinarow game on m×n board.
m,n,kgames are mainly of mathematical interest. One seeks to find the gametheoretic value, which is the result of the game with perfect play. This is known as solving the game.
Contents
Strategy stealing argument
A standard strategy stealing argument from combinatorial game theory shows that in no m,n,kgame can there be a strategy that assures that the second player will win (a secondplayer winning strategy). This is because an extra stone given to either player in any position can only improve that player's chances. The strategy stealing argument assumes that the second player has a winning strategy and demonstrates a winning strategy for the first player. The first player makes an arbitrary move to begin with. After that, he or she pretends that he or she is the second player and adopts the second player's winning strategy. He or she can do this as long as the strategy doesn't call for placing a stone on the 'arbitrary' square that is already occupied. If this happens, though, he or she can again play an arbitrary move and continue as before with the second player's winning strategy. Since an extra stone cannot hurt him or her, this is a winning strategy for the first player. The contradiction implies that the original assumption is false, and the second player cannot have a winning strategy.
This argument tells us nothing about whether a particular game is a draw or a win for the first player. Also, it does not actually give a strategy for the first player.
Applying results to different board sizes
If an (m,n,k) game is a draw, a smaller game (that is, one with lower m and n) is also a draw. This means that if an (m,n,k) game is a win, then any larger game must also be a win.
A useful notion is a "weak (m,n,k) game", where kinarow by the second player does not end the game with a second player win.
If weak (m,n,k) is a draw, then any smaller (in m and n) normal and weak (m,n,k) games are also a draw.
Conversely, if weak or normal (m,n,k) is a win, then any larger weak (m,n,k) is a win.
Note that proofs of draws using pairing strategies also prove a draw for the weak version and thus for all smaller versions.
General results
The following statements refer to the first player, assuming that both players use an optimal strategy.
 k ≥ 9 is a draw: when k = 9 and the board is infinite, the second player can draw via a "pairing strategy". A draw on an infinite board means that the game will go on forever with perfect play. A pairing strategy involves dividing all the squares of the board into pairs in such a way that by always playing on the pair of the first player's square, the second player is ensured that the first player cannot get k in a line. A pairing strategy on an infinite board can be applied to any finite board as well  if the strategy calls for making a move outside the board, then the second player makes an arbitrary move inside the board.^{[citation needed]}
 k ≥ 8 is a draw on an infinite board. It is not clear if this strategy applies to any finite board sizes.^{[2]} It is not known if the second player can force a draw when k is 6 or 7 on an infinite board.
 k ≥ 3 and either k > m or k > n is a draw, also by a pairing strategy in the dimension not smaller than k (or trivially impossible to win if both are smaller)^{[citation needed]}
 (k,k,k) is a draw if and only if k ≥ 2 (pairing strategies for k ≥ 6, special cases otherwise)^{[citation needed]}
Specific results
 k = 1 and k = 2 are trivial wins, except for (1,1,2) and (2,1,2)
 k = 3 is a draw for (3,3,3) (see Tictactoe) or if m < 3 or n < 3. It is a win otherwise.
 (m,4,4) is a win for m = 30 (Lustenberger, 1967) and a draw for m ≤ 8.^{[1]} In 2003, it was shown to be a win for m = 9 (Sobotovych, see external link W.J. Ma).
 (5,5,4) is a draw.
 (6,5,4) is a win.
 (6,6,5) is a draw.
 Computer search by L. Victor Allis has shown that (15,15,5) is a win, even with one of the restrictive rules of Gomoku.
Multidimensional variant
It is possible to consider variants played on a multidimensional instead of a bidimensional board.
For the case of kinarow where the board is an ndimensional hypercube with all edges with length k , Hales and Jewett proved^{[3]} that the game is a draw if k is odd and k ≥ 3^n  1 or if k is even and k ≥ 2^(n+1)  2.
They conjecture that the game is a draw also when the number of cell is at least twice the number of lines, which happens if and only if 2 k^n ≥ (k + 2)^n.
See also
References
 ^ ^{a} ^{b} J. W. H. M. Uiterwijk and H. J van der Herik, The advantage of the initiative, Information Sciences 122 (1) (2000) 4358.
 ^ ^{a} ^{b} H. Jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck (2002). "Games solved: Now and in the future". Artificial Intelligence.
 ^ Elwyn R. Berlekamp, John Horton Conway, Richard K. Guy. "Winning ways for your mathematical plays, Volume 3", A K Peters (2003)
External links
 W.J. Ma, Generalizations of tictactoe, [1].
Categories: Abstract strategy games
 Tictactoe
Wikimedia Foundation. 2010.
Look at other dictionaries:
MNK Fjaka — (FC Fjaka) is a football club in the city of Bol, Croatia, with its home in the Elaphusa stadium. Fjaka kit is orange black shirts and black shorts.HistoryThe club was founded by Nikola Ivelić and his teammates in summer 2005 in their hometown… … Wikipedia
Final Fantasy XI — Developer(s) Square Product Development Division 3 (pre April 1, 2003)[1] Square Enix Produ … Wikipedia
Character class (Dungeons & Dragons) — A character class is a fundamental part of the identity and nature of characters in the Dungeons Dragons role playing game. A character s capabilities, strengths, and weaknesses are largely defined by his or her chosen class; choosing a class is… … Wikipedia
Final Fantasy XI — Desarrolladora(s) Squaresoft Distribuidora(s) PlayStation 2 … Wikipedia Español
Carlsruhe — Karlsruhe Pour la ville des États Unis, voir Karlsruhe (Dakota du Nord). 49°00′N 08°24′E / … Wikipédia en Français
Karlsruhe — Pour la ville des États Unis, voir Karlsruhe (Dakota du Nord). Karlsruhe … Wikipédia en Français