What are some 'solved' games where the outcome is determined by perfect play by both players?
A paper published on October 30, 2023
Solved game - Wikipedia
https://en.wikipedia.org/wiki/Solved_game#Solved_games
Games that have been certified as 'solved' so far include:
◆Awari
It is a type of traditional board game `` Mancala '' that has been played in Africa and Southeast Asia for a long time, and is characterized by its unique rules called ``Grand Slam.'' In 2002, Henri Bal and John Roman of the Vrije Universiteit Amsterdam in the Netherlands reported that ``both players can force the game to a draw.''
◆Disposable chopsticks
``Disposable Chopsticks'' is a hand game in which ``Each player extends one finger and takes turns touching that finger to change the number of chopsticks.'' The game ends when both players play perfectly. I know that it will continue indefinitely.
◆Four-tac-toe (Connect Four)
``Four-tac-toe'' is a game in which two players take turns stacking pieces from the bottom, and the first person to line up four pieces vertically, horizontally, or diagonally in a straight line wins. This game was proven in 1988 by programmers James Allen and Victor Allis that ``if both players make the best move, the first player will always win'' (PDF file) .
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◆Fanorona
Fanorona , a classic board game (PDF file) , is a game where players use 22 pieces each on a 5 x 9 board to capture the pieces of their opponent. In 2008, Maarten Schad of Maastricht University and his colleagues reported (PDF file) that ``if both parties make their best moves, there will always be a draw.''
◆Gomoku
In Gomoku, a traditional Japanese board game played using Go equipment, players take turns placing stones on the board, and the first player to line up five of their own stones wins. In 1899, Ruika Kuroiwa discovered a winning strategy for Gomoku, and in 1993, Victor Aris proved that ``If you keep making the best moves, the first player will always win.''
◆Ghost
'Ghost' is a word game played by multiple players, with each player adding letters one letter at a time, and once the letters form a word, that player loses . Alan Frank, a member of the National Puzzlers' League, created a winning method for 'Ghost' in 1987.
◆Hexa Pawn
The game is played on a 3x3 square chess board, and each player uses three pieces to ``advance their piece to the third square in front'', ``take all of the opponent's pieces'', and ``take the opponent's pieces.'' You will win if you accomplish either of the following. It has become clear that when both players make the best move, the second mover always wins.
The 'L game' is a game where you move 3x2 L-shaped blocks or 1x1 pieces to each other on a 4x4 grid to eliminate the destination of your opponent's L-shaped block. This game has been proven through various experiments that the best move always results in a draw.
◆Losing Chess
Although the rules are the same as chess, the victory conditions are opposite to those of general chess, such as ``losing all pieces'' and ``reaching a deadlock''. In this game, if you can take the opponent's piece, you must take it, so if you keep making the best moves, the first player will always win, as reported by Mark Watkins of the University of Sydney (PDF file). Masu.
◆Maharaja and Sepoys
``Maharaja and Sepoy'' is a game where one side is a general chess layout and the other side is a battle with only one king piece that can move as a queen or a knight. In this game, the winning condition is to checkmate each other's kings. So far, it has been proven that when both players make the best move, the player who moves one piece always wins.
In ``Nim'', two players compete for coins from several piles, and the player who takes the last coin wins, and there are established winning methods for both the first move and the second move.
◆Nine Men's Morris
``Nine Men's Morris'' is a game in which players compete by moving pieces along a specified line, and the winner is the one who reduces the opponent's pieces to two or makes them unable to move. Ralph Gasser proved in 1993 that ``either player can bring the game to a draw'' ( PDF file) .
◆Order and chaos
'Chaos and Order' is a game similar to Gomoku, which is played using two types of pieces. In this game, it is clear that if you make the best move, the first player will always win.
◆Ovalf
One type of Mancala developed in Southeast Asia is 'Ovalhu'. So far, computers have discovered winning methods.
◆Panki
``Punki'' is a game that developed as a type of ``Sink Cummins'' where you move five pieces on a 5 x 5 grid. Punki is different from Sink Cummins in that it moves six pieces on a 4x4 square, and it has been reported that if you take the best move, it will always result in a draw.
◆Pentago
'Pentago' is a game similar to Gomoku that is played using four 3x3 boards. In this game, the winning condition is to see who can line up 5 balls while rotating the board. It has been proven that if both players play perfectly, the first player always wins.
◆Pentomino
This is a game that uses pieces of various shapes made up of 5 square blocks, and takes turns placing pieces in an 8 x 8 square, and the player who cannot place any more loses. In 'Pentomino', it is clear that if both players do their best, the first player will always win.
◆Quarto
'Quarto' is a game in which the player who lines up four pieces in a row vertically, horizontally, and diagonally on a 16-square 4x4 board wins. Unlike regular 4-tac-toe, you need to match the pieces in a row based on some characteristic such as color, shape, height, or presence or absence of holes. This game has also been proven to result in a draw if both players make their best moves.
◆Cubic
'Cubic' is a 4x4x4 three-dimensional tic-tac-toe game. In this game, computer scientist Oren Patashnik and others have shown that if you make the best move, the first player will always win.
◆Renju
Renju is a board game that adds various rules to Gomoku to make it more competitive. This game was created in an attempt to change Gomoku, which gives the first player an advantage, by adding multiple rules, but János Wagner and others have reported that if they continue to make the best moves, the first player will always win.
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◆Sim
Each player takes turns coloring one line connecting the vertices of the hexagon. The goal is to keep coloring so that you don't end up with a triangle made up only of your own color. It has been shown that when both players do their best, the last player wins.
◆Teaco
'Teiko' is a game with rules similar to 4-tac-toe, where you can move pieces between each other. In 1998, computer scientist Guy Steele proved that if both players played perfectly, the first player would win or draw.
◆Three Men Morris
``Three Men's Morris'' is a game in which the player who moves his pieces to each other on a 3x3 board wins by arranging three of his pieces. It has become clear that if we give our best to each other, there will always be a draw.
◆Three Musketeers
The players are divided into either the Three Musketeers, who use only three pieces, or the enemy, who uses 22 pieces, and move pieces with each other on a 5x5 board. The enemy wins by arranging all the three musketeers in the same row. The Three Musketeers win when there are no enemy pieces adjacent to them and they can no longer move. Mr. Ali Elabridi has proven (PDF file) that 'if you do your best, the enemy will always win.'
◆Tic-tac-toe
It's a simple game where you draw O's and O's on each other on a 3x3 grid, and the first person to line up 3 wins. It has become clear that if both sides give their best, a draw is bound to result.
◆Tiger and goat
The main purpose of the tiger side is to hinder the actions of the goat side, which uses 4 pieces to move up to 20 pieces. Yu Jin Lim has proven (PDF file) that even in the best of circumstances, a draw will always result.
◆Wysof's game
Remove coins from the two piles of coins in order. It is possible to remove coins from one or both piles, but you must remove the same number from both piles. The player who takes the last coin wins. A winning method has been discovered by Dutch mathematician Willem Wysoff.
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