Chopsticks (hand game)

1. Each player begins with one finger raised on each hand. After the first player, turns proceed clockwise.
2. On a player's turn, they must either attack or split, but not both.
3. To attack, a player uses one of their live hands to strike an opponent's live hand. The number of fingers on the opponent's struck hand will increase by the number of fingers on the hand used to strike.
4. To split, a player strikes their own two hands together, and transfers raised fingers from one hand to the other as desired. A move is not allowed to simply reverse one's own hands (e.g. going [2, 3]–>[3, 2] is prohibited) and they must not kill one of their own hands (e.g. going [1, 1]–>[0, 2] is prohibited).
5. If any hand of any player reaches exactly five fingers, then the hand is killed and this is indicated by raising zero fingers (i.e. a closed fist). A player may revive their own dead hand using a split, as long as they follow the rules in #4. However, players may not revive opponents' hands using an attack. Therefore, a player with two dead hands can no longer play and is eliminated from the game.
6. If any hand of any player reaches more than five fingers, then five fingers are subtracted from that hand. For instance, if a 4-finger hand strikes a 2-finger hand, then the 2-finger hand will have 6 fingers. 5 fingers will be subtracted leaving one remaining. This is called roll-over.
7. A player wins once all opponents are eliminated (by each having two dead hands at once).
8. There is also a variation in which a player can kill their own hand.

Using the rules above, two perfect players will play indefinitely, going in a loop. In fact, even very inexperienced players can avoid losing by simply looking one move ahead.

Using the cut-off variant, the first player can force a win. One winning strategy is to always reach one of the following configurations after each of your moves, preferentially choosing the first one in the list if there is more than one choice. Each configuration will be given as [a,b],[c,d] where [a,b] represents your two hands (ignoring order) and [c,d] represents your opponent's.

• [2,1],[1,1] (You start here.)
• [?,?],[1,2] (Win immediately if possible.)

Conversely, if tapping one's own hand is not allowed, but splitting two live hands into one is allowed, then the second player has a winning strategy.[3]

A chopsticks position can be easily abbreviated to a four-digit code [ABCD]. A and B are the hands (in ascending order of fingers) of the player who is about to take their turn. C and D are the hands (in ascending order of fingers) of the player who is not about to take their turn. It is important to notate each player's hands in ascending order, so that a single distinct position isn't accidentally represented by two codes. For example, the code [1032] is a mistake, and should be notated [0123].

Therefore, the starting position is [1111]. The next position must be [1211]. The next position must be either [1212] or [1312]. Treating each position as a 4-digit number, the smallest position is 0000, and the largest position is 4444.

This abbreviation formula expands easily 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).

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 less of these are available on a given turn. For example, the early position 1312 can go to 2213, 1313, 2413, 0113, or 1222.

• Suicide: You are allowed to kill one of your own hands with a split. This allows for quicker defeations. For example, in the position [1201], you could execute 12-03, thus bringing the game to [0103]. The opponent is forced to play B-D, bringing the game to [0401], at which point you could swiftly finish them off.
• Swaps: If you have two unequal live hands, you may swap them (essentially forfeiting your turn).
• Sudden Death: You lose when you only have one finger left total. Alternately, each player could begin with three lives, and every time they get down to [01], they lose a life.
• Meta: If your hands add up to over five, you can combine them, subtract five from the total, and then split up the remainder. For example, [44] adds up to 8. So under Meta rules, you can combine them into 8, which becomes 3, which you could then split into [12]. Therefore you could go from [44] to [12] in a single move. Meta unlocks 2 new possible moves (34-11, 44-12). If playing both Meta and Suicide, then 4 additional moves are unlocked (24-01, 33-01, 34-02, 44-03), for a total maximum of 20 possible moves.
• Logan Clause: You are allowed to suicide and swap, but only if you do both at the same time (i.e. swap your dead hand for your live one).
• Cutoff: If a hand gets above 5 fingers, it is dead. (As opposed to rollover, as described in the official rules)
• Zombies: In 3 or more players, if a player is knocked out, then he is permanently reduced to 1 finger on 1 hand. On his turn, he may attack, but he may not split or be attacked (invented by Chris Bandy).
• Halvesies: Splitting is only allowed if you are dividing an even number into two equal halves. Or optionally, an odd number being divided as evenly as possible (using whole numbers). In this variation the second player has a winning strategy (can always force a win).[4]
• Stumps: If you are at [01], you are allowed to split to [0.5 0.5].
• Larger Numbers: Numerals from 1–9 are allowed, where a hand dies at 10. In this case, Chinese hand numerals are often used. This variation often includes roll-overs.
• Suns: Both players start with a 4 in each of their hands ([4444]). This is a position that is unreachable in normal gameplay (i.e. from the opening position [1111]).
• Integers: You are allowed to swap one of your own hands by flipping it over, changing the +/- sign of the hand. This allows for negative and 0 value hands, though a hand still dies at 5 or -5.

Since the roll-over amount is 5, chopsticks is a base-5 game. Each position is four digits long. Counting from 0000 to 4444 (in base-5) gives us 625 positions. However, most of these positions are incorrect notations (e.g. 0132, 1023, and 1032). They appear different but are functionally the same in gameplay. To find the number of functionally distinct positions, we simply square the number of functionally distinct pairs. There are 15 distinct pairs (00, 01, 02, 03, 04, 11, 12, 13, 14, 22, 23, 24, 33, 34, and 44). Since either player could have any of these pairs, we simply multiply 15*15, which gives us 225 functionally distinct positions.

• There are 625 positions, including redundancies.
• There are 225 functionally distinct positions.
• There are 204 reachable positions.

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. Fifteen of these 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 you can't lose on your own turn, these 15 positions are obviously unreachable. The remaining six positions are not as obviously unreachable, but can be demonstrated as such:

1. 1101 is unreachable because the player who just went [01] would not be able to split, so therefore that player must have attacked using his [01]. But there's no way to use [01] to attack an enemy so that they move to [11]. That would require attacking a dead hand, which is illegal.
2. 2202 is unreachable for the same reasons as 1101. [02] cannot be a product of a split, since killing your own hand is illegal. And using a 2 to attack an enemy, so that the enemy's pair becomes [22], is impossible, as that again would require attacking a dead hand.
3. 3303 is unreachable for the same reasons as 1101 and 2202.
4. 3444 is actually reachable, but only from 4444. Since 4444 is not reachable from 1111, neither is 3444.
5. 4404 is unreachable for the same reasons as 1101, 2202, and 3303.
6. 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.

In the "Suns" variant, two new positions are unlocked (4444, 3444), for a total of 206 reachable positions. However, after two moves, it is impossible to return to either 4444 or 3444.

In the "Suicide" variant, three new positions are unlocked (2202, 3303, 4404), for a total of 207 reachable positions. Unlike the "Suns" variant, these three positions can be accessed mid-game.

Therefore, playing "Suicide" and "Suns" together unlocks five new positions, for a total of 209 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 amount of moves.

Chopsticks can be generalized into a (p,h,r)-type game, where p is the number of players, h is the number of hands each player has, and r is the roll-over amount. In addition, players can start with varying numbers of sticks in each hand, although this isn't an essential property of the game.

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

• With 2 hands, there are 204 positions.
• With 3 hands, there are 1,183 positions.
• With 4 hands, there are over 4,000 positions.

• With a roll-over of 5, there are 204 positions.
• With a roll-over of 6, there are 412 positions.
• With a roll-over of 7, there are 748 positions.
• With a roll-over of 8, there are 1,250 positions.
• With a roll-over of 9, there are 1,969 positions.
• With a roll-over of 10, there are 2,958 positions.

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

1. 1111 1211 1312 0113 1401 0014

The longest possible game that gets farther from the starting point with each move is 9 moves. There are two instances:

1. 1111 1211 1212 2212 2322 0223 0202 0402 0104 0001
2. 1111 1211 1212 2312 2323 0323 0303 0103 0401 0004

The longest possible game with revisitation is indefinite.

• Morra (game) – a different hand-game, which is based on chance rather than logic.

• Unbeatable Chopsticks AI Bot. This bot plays the game with remainders and transfers.