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Source: Chopsticks (hand game)

Chopsticks (sometimes called Calculator, or just Sticks) 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. 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.

Description

= Gameplay

Chopsticks

= Abbreviation

Chopsticks

= Moves

= Game lengths

= Positions

Chopsticks

. 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.

Generalisations

Chopsticks can be generalized into a

(
p
,
r
)

{\displaystyle (p,r)}

-type game, where

p

{\displaystyle p}

is the number of players and

r

{\displaystyle r}

is the rollover amount.

= 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

p
=
2

{\displaystyle p=2}

and a rollover of

r

{\displaystyle r}

,

There are

r

2
p

=

r

4

{\displaystyle r^{2p}=r^{4}}

positions, including redundancies.
There are

(

r
+
1

2

)

{\displaystyle {r+1 \choose 2}}

distinct finger pairs (the

r

{\displaystyle r}

-th triangular number), and thus

(

r
+
1

2

)

p

{\displaystyle {r+1 \choose 2}^{p}}

functionally distinct positions.
For

r
>
2

{\displaystyle r>2}

, there are

(

r
+
1

2

)

+
(
r
−
1
)
+
2

{\displaystyle {r+1 \choose 2}+(r-1)+2}

unreachable positions.

(

r
+
1

2

)

{\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.

(
r
−
1
)

{\displaystyle (r-1)}

unreachable positions occur when the current player has

A
=
B
=
k

{\displaystyle A=B=k}

and the other player has

C
=
0

{\displaystyle C=0}

and

D
=
k

{\displaystyle D=k}

, for

0
<
k
<
r

{\displaystyle 0
. 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

k

{\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

r
−
1

{\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

r
−
1

{\displaystyle r-1}

would have required attacking a dead hand, which is illegal.
An unreachable position occurs when the current player has

A
=
r
−
2

{\displaystyle A=r-2}

and

B
=
r
−
1

{\displaystyle B=r-1}

, and the other player has

C
=
D
=
r
−
1

{\displaystyle C=D=r-1}

. This position has only one previous position, which is unreachable from the starting position.
Thus, for

r
>
2

{\displaystyle r>2}

, there are

(

r
+
1

2

)

p

−

(

(

r
+
1

2

)

+
(
r
−
1
)
+
2

)

{\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

2

p

−
1

{\displaystyle 2^{p}-1}

reachable positions in the game, and a game length of

2
p
−
1

{\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

r

{\displaystyle r}

, each position after

k

{\displaystyle k}

moves in the game can be represented by the tuple

(

F

k
+
2

mod

r

,

F

k
+
1

mod

r

)

{\displaystyle \left(F_{k+2}{\bmod {r}},F_{k+1}{\bmod {r}}\right)}

, where

F

k

{\displaystyle F_{k}}

is the

k

{\displaystyle k}

-th Fibonacci number with

F

0

=
0

{\displaystyle F_{0}=0}

and

F

1

=
1

{\displaystyle F_{1}=1}

. The number of positions is given by least positive number

k

{\displaystyle k}

such that

r

{\displaystyle r}

divides

F

k
+
2

{\displaystyle F_{k+2}}

. This variant is strongly solved as a win for either side depending upon

r

{\displaystyle r}

and the divisibility properties of Fibonacci numbers. The length of the game is

k
+
1

{\displaystyle k+1}

.

Variations

Misère: First player to have both of their hands killed wins.
Suicide: Players are allowed to kill one of their own hands with a split. For example, in the position 1201, a player 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 a quick win for the first player is possible.
Swaps/Cherri: If players have two unequal live hands, they may swap them (though forfeiting their turn). This variation commonly yields a draw by repetition or infinite loop for obvious reasons. To avoid this, limits can be placed on the number of consecutive swaps a player can do without being attacked before they are forced to attack.
Sudden Death: Players lose when they only have one finger left (on both hands). Alternately, each player could begin with three lives, and every time they get down to [01], they lose a life.
Meta: If a player's hands add up to over five, they can combine them, subtract five from the total, and then split up the remainder. For example, [44] adds up to 8, which becomes 3 after subtracting five; these can then be split into [12]. Therefore, it is possible to 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, four additional moves are unlocked (24→01, 33→01, 34→02, 44→03), for a maximum of 20 possible moves in total.
Logan Clause: Players are allowed to suicide and swap, but only if doing both simultaneously (i.e. swapping a dead hand for a live one).
Death Attack: Players are allowed to attack dead hands. Attacking a dead hand will treat the dead hand as a 0 before being attacked.
Cutoff or Game of Five: If a hand gets above five fingers, it is dead (as opposed to rollover, described in the official rules).
Zombies: With three or more players, if a player is knocked out, then they are permanently reduced to one finger on one hand. On their turn, they may attack, but may not split or be attacked (invented by Chris Bandy).
Transfers only: Divisions are not allowed. The only splits allowed are transfers.
Divisions only: Transfers are not allowed. The only splits allowed are divisions.
Splits: Splitting is only allowed when 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).
Stumps / Knubs: If a player is at [01], it is legal to split into [0.5 0.5], represented by curling a finger down halfway.
More Hands: Each player has "more than two hands", usually represented by playing in teams of multiple people. With more hands per player, different transfer, division, swapping, and suicide rules are possible, including:
Single transfer: Each player can transfer fingers between only two hands.
Multiple transfer: Each player can transfer fingers between more than two hands, so long as the resulting state is different from the original state.
Single Division: The player can transfer fingers from only one hand to only one dead hand.
Partition: The player can transfer fingers from only one hand to multiple dead hands.
Transfer and partition: The player can transfer fingers from multiple hands to revive dead hands.
Different Numbers: A hand dies when it reaches a positive number

r

{\displaystyle r}

(in standard Chopsticks,

r
=
5

{\displaystyle r=5}

). Different hand counting systems could be used for numbers greater than 5 such as Chinese hand numerals, senary finger counting, and finger binary. This variation often includes rollovers.
Suns: Both players start with a 4 on each hand (4444). This is a position that is unreachable in normal gameplay (i.e. from the opening position 1111).
Integers: It is permitted to swap one of one's own hands by flipping it over, changing the +/- sign of the hand. This allows for negative and zero value hands, though a hand still dies at 5 or −5. With rollover, this action becomes identical to replacing the value of the hand with 5 minus the value.
Unnamed: Attacking own hands are allowed, adding two additional moves (A→B, B→A). Typically played in conjunction with Swap and Cutoff variants.
Cherry Bomb: Transferring to exactly 5 will cause the attacker's hand to become [11] and the target's hand to become [01]. In a two-player game, the position becomes 1101, which becomes an easy forced win, so it is best played with more than 2 people.
Self-adding: As their turn, players can add 1 finger to any living hand, or transfer multiple fingers to a dead hand. This enables self-destruction (by adding 1 to a living hand with 4 fingers) and resurrection. This game variation has more complicated strategy.

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.
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):

1211 (starting here)
AB12, where A and B can be any number of fingers (winning immediately if possible)
Conversely, in the Division and Suicide only variation, then the second player has a winning strategy.