Chopsticks (hand game)


Chopsticks is a hand game for two players in which players extend a number of fingers from each hand and transfer those scores by taking turns to tap one hand against another. 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.

Rules

  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 and they must not kill one of their own hands.
  5. If any hand of any player reaches exactly five fingers, then the hand is killed and this is indicated by raising zero fingers. A player may revive their own dead hand using a split, as long as they abide by the rules for splitting. 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. Five fingers will be subtracted, leaving one remaining. This is called roll-over.
  7. A player wins once all opponents are eliminated.
  8. There is also a variation in which a player can kill their own hand.

    Optimal strategy

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.
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 move, preferentially choosing the first one in the list if there is more than one choice. Each configuration will be given as , where represents a player's two hands and represents their opponent's.
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.

Abbreviation

A chopsticks position can be easily abbreviated to a four-digit code . A and B are the hands of the player who is about to take their turn. C and D are the hands 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 is a mistake, and should be notated .
Therefore, the starting position is . The next position must be . The next position must be either or . 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, 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.

Moves

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

Game lengths

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.

Positions

Since the roll-over amount is 5, chopsticks is a base-5 game. Each position is four digits long. Counting from 0000 to 4444 gives us 625 positions. However, most of these positions are incorrect notations. 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. Since either player could have any of these pairs, we simply multiply 15*15, which gives us 225 functionally distinct 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.
  1. 15 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. Since the player can't lose on their own turn, these positions are obviously unreachable.
  2. 4 of those pairs are where the player to move having , and the other player having , where. This is unreachable because the player who just went would not be able to split, so therefore that player must have attacked using his . But there's no way to use to attack an enemy so that they move to . That would require attacking a dead hand, which is illegal.
  3. The remaining two positions are 3444 and 4444. 4444 is unreachable because a player cannot reach from a split, and therefore had to already have . The only possible pair that goes to after being attacked by is , 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.
All but one of these positions in point 2 are reachable in the "Suicide" variant, as is still unreachable. is reachable if the "Suicide" variant is played with the "Meta" variant. The two positions in point 3 are reachable in the "Suns" variant, as 4444 is the starting position, but the two positions cannot be accessed mid-game. Therefore, if playing "Suicide", "Meta", and "Suns" together, there are a total of 15 unreachable positions and 210 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.

Variations

Chopsticks can be generalized into a -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.

Degenerate cases

A game with a roll-over amount of 1 is the trivial game, because all hands are dead at start as values of one become values of zero. A game with one or less players is not a game, but a puzzle or an cellular automaton.
A game with a roll-over amount of 2 is degenerate, because splitting is impossible and the roll-over and cutoff variations result in the same game. Hands are either 'alive' and 'dead', and attacking a hand kills the hand. In fact, one could simply keep count of the number of 'hands' a player has, and when a player attacks an opponent, the number of hands that opponent has decreases by one. There are a total of reachable positions in the game, and a game length of. The two player game is strongly solved as a first person win for any. Playing this degenerate variant with the "Stumps" variant yields a game that is isomorphic to a "Halvesies" variant with a roll-over amount of 4 and a starting position where all players have two fingers on every hand.

Two players

When each player has only one hand, the game becomes degenerate, because splits cannot occur and each player only has one move. Given a roll-over of each position after moves in the game can be represented by the tuple, where is the -th Fibonacci number with and. The number of positions is given by least positive number such that divides. This variant is strongly solved as a win for either side depending upon and the divisibility properties of Fibonacci numbers. The length of the game is.
When each player has more than one hand, each hands, given a roll-over of,
Since the roll-over amount is, chopsticks is a base- game. Each position is digits long. Enumerating all numbers in base- with digits gives us positions. However, most of these positions are incorrect notations. They appear different but are functionally the same in gameplay. To find the number of functionally distinct positions, we square the number of functionally distinct pairs. For a roll-over of and hands, there are distinct pairs, where is the -th -simplex number. Since either player could have any of these pairs, we simply square the resulting value, which gives us functionally distinct positions.
There are unreachable positions.
  1. of these are simply one player having each of the distinct pairs, and the other player being dead. The problem is that the dead player is the player who just took his turn. Since the player can't lose on their own turn, these positions are obviously unreachable.
  2. of those positions are when the player, whose turn it is, has hands of value for and the other player has only one alive hand of value. These positions are unreachable because the player who only has one alive hand of value would not be able to split, so therefore that player must have attacked using his sole alive head. But there is no way to use his sole alive hand to attack an enemy so that they have hands of value, as that would require attacking a dead hand, which is illegal.
  3. of those positions are when the player, whose turn it is, has hands of value and the other player has alive hand of value, where. These position are unreachable because any player who only has hands of value would not be able to split, so therefore that player must have attacked using one of his value hands. But there is no way to use an valued hand so that they have hands of value, as that would require attacking a dead hand, which is illegal.
  4. The position when both players have hands of value. This is unreachable for the same reason as point 3 above.
  5. The position when the player whose turn it is has one hand of value and hands of value, and the other player has hands of value. This position is only reachable from the previous position, but the previous position is not reachable from the starting position, so this position isn't either.
All but one of these positions in points 2 and 3 are reachable in the "Suicide" variant, as the position where the player whose turn it is has hands of value 1 and the other player has only one alive hand of value 1 is still unreachable. That position is reasonable is reachable if the "Suicide" variant is played with the "Meta" variant. The two positions in points 4 and 5 are reachable in the "Suns" variant, as the position in point 4 is the starting position, but the two positions cannot be accessed mid-game. Therefore, if playing "Suicide", "Meta", and "Suns" together, there are a total of unreachable positions and reachable positions.
HandsRoll-over amountPositionsFunctionally Distinct PositionsReachable PositionsWith 'Suicide', 'Meta', and 'Suns'
2381362630
337291008590
436561225204210
5359049441413420
63531441784748756
242561008590
344096400374380
4465536122511831190
541048576313630723080
6416777216705669636972
25625225204210
3515625122511831190
45390625490048224830
559765625158761574115750
65244140625441004388043890

More than two players

Given a roll-over of 5 and 2 hands.