Gameplay

Chopsticks consists of players tallying points using the fingers on their hands.

Each player starts with two points (one finger on each hand). Taking turns, players tap the opponent's hand, which adds points to it equal to the value of the tapping hand. A player’s hands do not change when the opponent’s hand is tapped. For example, if an opposing player has the maximum number of points on their hand, they may not subtract points from it if they decide to knock out the other player's hand, such as if a player has five points and the other has two, the player with five points cannot give the other player a portion of their points to avoid being knocked out. When a hand gets five or more points, it is "knocked out" and called a dead hand.^[3] A dead hand cannot attack or be attacked. A player wins by knocking out both of their opponent's hands.

Instead of attacking on their turn, a player may "split" points among their hands. A split can be either a transfer or a division. A transfer involves moving a certain number of points from one living hand to another; transferring all points off one hand knocks it out ("suicide"), and is allowed in some variations.^[3] A division can resurrect a dead hand by moving points from a living hand, bringing it back into play.^[3] The new distribution must be distinct from the original distribution; a player may not simply swap points between hands.^[3]

Due to the game's simple basic structure, there are many variations with additional rules.^[1] In some variations, a sum greater than 5 "rolls over" to a smaller value by subtracting 5 from the sum (modular arithmetic); a hand is eliminated only when it has exactly five points. In other variations, more complex transfer and division moves are allowed.

Abbreviation

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. For example, the code 1023 is not allowed, and should be notated 0123.

The starting position is 1111. The next position must be 1211. The next position must be either 1212 or 1312. During the game, the smallest position is 0000, and the largest position is 4444.

This abbreviation can be expanded to games with more players. 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).

Moves

Under normal rules, there are a maximum of 14 possible moves:

• Four attacks (A→C, A→D, B→C, B→D)
• Four divisions (02→11, 03→12, 04→13, 04→22)
• Six transfers (13→22, 22→13, 14→23, 23→14, 24→33, 33→24)

However, only 5 or fewer of these are available on a given turn. For example, the early position 1312 can become 2213, 1313, 2413, 0113, or 1222.

Game lengths

The shortest possible game is 5 moves. There is one instance:

• 1111 1211 1312 0113 1401 0014

Without revisitation (repeating a position), the longest possible game is 9 moves. There are two instances:

• 1111 1211 1212 2212 2322 0223 0202 0402 0104 0001
• 1111 1211 1212 2312 2323 0323 0303 0103 0401 0004

With revisitation, the longest possible game is indefinite.

Positions

Since the roll-over amount is 5, Chopsticks is a base-5 game. In a two-player game, each position is four digits long. Counting from 0000 to 4444 (in base 5) yields 625 positions. However, this includes redundancies—most of these positions are incorrect notations (e.g. 0132, 1023, and 1032 are incorrect notations of 0123), which appear different but are functionally the same in gameplay.

To find the number of functionally distinct positions, note that each player can be one of 15 distinct pairs (00, 01, 02, 03, 04, 11, 12, 13, 14, 22, 23, 24, 33, 34, and 44). With two players, there are 15*15 = 225 functionally distinct positions. In general, for ${\displaystyle n}$ players, there are ${\displaystyle 15^{n}}$ functionally distinct positions.

However, there are 21 unreachable positions: 0000, 0100, 0200, 0300, 0400, 1100, 1101, 1200, 1300, 1400, 2200, 2202, 2300, 2400, 3300, 3303, 3400, 3444, 4400, 4404, and 4444.

• 15 of these positions are simply one player having each of the 15 distinct pairs, and the other player being dead. The problem is that the dead player is the player who just took his turn (hence the "00" on the right side). Since the player can't lose on their own turn, these positions are unreachable.
• 4 of these positions are the player to move having [kk], and the other player having [0k], where ${\displaystyle 0<k<5}$. This is unreachable because the player who just went [0k] would not be able to split, so therefore that player must have attacked using his [0k]. But there's no way to use [0k] to attack an enemy so that they move to [kk]. That would require attacking a dead hand, which is illegal.
• The remaining two positions are 3444 and 4444. 4444 is unreachable because a player cannot reach [44] from a split, and therefore had to already have [44]. The only possible pair that goes to [44] after being attacked by [44] is [04], which again requires that a dead hand be attacked. 3444 is actually reachable, but only from 4444. Since 4444 is not reachable from 4444, neither is 3444.

This gives a total of 204 unique, reachable positions.

There are 14 reachable endgames: 0001, 0002, 0003, 0004, 0011, 0012, 0013, 0014, 0022, 0023, 0024, 0033, 0034, 0044. 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.

Fewer than two players

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.

Two players

Given ${\displaystyle p=2}$ and a rollover of ${\displaystyle r}$,

• There are ${\displaystyle r^{2p}=r^{4}}$ positions, including redundancies.
• There are ${\displaystyle {r+1 \choose 2}}$ distinct finger pairs (the ${\displaystyle r}$-th triangular number), and thus ${\displaystyle {r+1 \choose 2}^{p}}$ functionally distinct positions.
• For ${\displaystyle r>2}$, there are ${\displaystyle {r+1 \choose 2}+(r-1)+2}$ unreachable positions.
  • ${\displaystyle {r+1 \choose 2}}$ unreachable positions occur when the current player has any distinct pair and the other player is dead. However, the dead player is the player who just took his turn. Since a player can't lose on their own turn, these positions are unreachable.
  • ${\displaystyle (r-1)}$ unreachable positions occur when the current player has ${\displaystyle A=B=k}$ and the other player has ${\displaystyle C=0}$ and ${\displaystyle D=k}$, for ${\displaystyle 0<k<r}$. These positions are unreachable because the other player [CD] did not have split on the previous turn, so must have attacked using their alive hand. However, there is no way to attack as such that results in the target having both hands of value ${\displaystyle k}$, as this would require attacking a dead hand, which is illegal.
  • An unreachable position occurs when both players have two hands of value ${\displaystyle r-1}$. As such, the previous player could not have split, so must have attacked. But to result in the opponent's hands both having of value ${\displaystyle r-1}$ would have required attacking a dead hand, which is illegal.
  • An unreachable position occurs when the current player has ${\displaystyle A=r-2}$ and ${\displaystyle B=r-1}$, and the other player has ${\displaystyle C=D=r-1}$. This position has only one previous position, which is unreachable from the starting position.

Thus, for ${\displaystyle r>2}$, there are ${\displaystyle {r+1 \choose 2}^{p}-\left({r+1 \choose 2}+(r-1)+2\right)}$ reachable positions.

More than two players

Given a rollover of 5,

• With 2 players, there are 204 positions.
• With 3 players, there are 3,337 positions.
• With 4 players, there are over 25,000 positions.

Degenerate cases

A game with a rollover amount of 1 is the trivial game, because all hands start dead.

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. Hands are either alive and dead, with no middle state, and attacking a hand kills the hand. 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. There are a total of ${\displaystyle 2^{p}-1}$ reachable positions in the game, and a game length of ${\displaystyle 2p-1}$. The two player game is strongly solved as a first person win.

When two players have only one hand, the game becomes degenerate, because splits cannot occur and each player only has one move. Given a rollover of ${\displaystyle r}$, each position after ${\displaystyle k}$ moves in the game can be represented by the tuple ${\displaystyle \left(F_{k+2}{\bmod {r}},F_{k+1}{\bmod {r}}\right)}$, where ${\displaystyle F_{k}}$ is the ${\displaystyle k}$-th Fibonacci number with ${\displaystyle F_{0}=0}$ and ${\displaystyle F_{1}=1}$. The number of positions is given by least positive number ${\displaystyle k}$ such that ${\displaystyle r}$ divides ${\displaystyle F_{k+2}}$. This variant is strongly solved as a win for either side depending upon ${\displaystyle r}$ and the divisibility properties of Fibonacci numbers. The length of the game is ${\displaystyle k+1}$.