Chopsticks (hand game)
Splits (sometimes called Calculator, Chopsticks, or just Sticks)[citation needed] is a hand game for two or more players, in which players extend a number of fingers from each hand and transfer those scores by taking turns tapping one hand against another.
[1][2] Chopsticks is an example of a combinatorial game, and is solved in the sense that with perfect play, an optimal strategy from any point is known.
[3] In Chopsticks, players tally points using the fingers of both hands, with each extended finger counting as one point.
[2][3][4] In the basic game for two players, each starts with two points—with one finger of each hand extended.
Players then take turns, and on each turn must either: (1) "Attack” by tapping either of the opponent's hands, adding points to it.
Due to the game's simple basic structure, there are many rule variations, as noted below.
In the "suicide" variation, a player may knock out their own hand by transferring all points from it, reducing it to zero.
In other variations, and for multi-player or team play, more complex transfer and division moves are allowed.
Each position in a two-player game of Chopsticks can be encoded as a four-digit number, with each digit ranging from 0 to 4, representing the number of active fingers on each hand.
This can be notated as [ABCD], where A and B are the hands of the player who is about to take their turn, and C and D are the hands of the player who is not about to take their turn.
Each pair of hands is notated in ascending order, so every distinct position is represented by one and only one four-digit number.
A three-player game can be represented by six digits (e.g. [111211]), where each pair of adjacent digits represents a single player, and each pair is ordered based on when players will take their turns.
The leftmost pair represents the hands of the player about to take his turn; the middle pair represents the player who will go next, and so on.
The rightmost pair represents the player who must wait the longest before his turn (usually because he just went).
Under normal rules, there are a maximum of 14 possible moves: However, only 5 or fewer of these are available on a given turn.
There is one instance: Without revisitation (repeating a position), the longest possible game is nine moves.
There are two instances: With revisitation, the longest possible game is indefinite.
In a two-player game, each position is four digits long.
Satisfyingly enough, these are all the 14 possible endgames; in other words, someone can win using any of the 14 distinct live pairs.
Out of these 14 endgames, the first player wins 8 of them, assuming that the games are ended in the minimum number of moves.
[further explanation needed] In a one-player game, the player trivially wins for virtue of being the last player in the game.
A game with zero players is likewise trivial as there can be no winners.
A game with a rollover amount of 2 is degenerate, because splitting is impossible, and the rollover and cutoff variations result in the same game.
In fact, one could simply keep count of the number of 'hands' a player has (by using fingers or some other method of counting), and when a player attacks an opponent, the number of hands that opponent has decreases by one.
The two player game is strongly solved as a first person win.
This variant is strongly solved as a win for either side depending upon
Using the rules above, two perfect players will play indefinitely; the game will continue in a loop.
In fact, even very inexperienced players can avoid losing by simply looking one move ahead.
In the cutoff variation, the first player can force a win.
One winning strategy is to always reach one of the following configurations after each move (preferentially choosing the first one): Conversely, in the Division and Suicide only variation, then the second player has a winning strategy.[5][how?]
