Chopsticks (hand game)

Chopsticks 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 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.
This official set of rules is called "rollover" where five fingers are subtracted should a hand's sum exceeds 5 as described below.
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 [a, b], [c, d] where [a,b] represents a player's two hands (ignoring order) and [c,d] 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 [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 not allowed, 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).
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:
The longest possible game that gets farther from the starting point with each move is 9 moves. There are two instances:
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 (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 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. 
All but one of these positions in point 2 are reachable in the "Suicide" variant, as [1101] is still unreachable. [1101] 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.
Generalisations.
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. 
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 a 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 (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 formula_4 reachable positions in the game, and a game length of formula_5. The two player game is strongly solved as a first person win for any formula_6. 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 (formula_7), the game becomes degenerate, because splits cannot occur and each player only has one move. Given a roll-over of formula_2 each position after formula_9 moves in the game can be represented by the tuple formula_10, where formula_11 is the formula_9-th Fibonacci number with formula_13 and formula_14. The number of positions is given by least positive number formula_9 such that formula_2 divides formula_17. This variant is strongly solved as a win for either side depending upon formula_2 and the divisibility properties of Fibonacci numbers. The length of the game is formula_19. 
When each player has more than one hand (formula_20), each hands, given a roll-over of formula_2,
Since the roll-over amount is formula_2, chopsticks is a base-formula_2 game. Each position is formula_27 digits long. Enumerating all numbers in base-formula_2 with formula_27 digits gives us formula_22 positions. However, most of these positions are incorrect notations (e.g. 001210, 010120, and 100021 for formula_31). 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 formula_2 and formula_6 hands, there are formula_34 distinct pairs, where formula_34 is the formula_2-th formula_6-simplex number. Since either player could have any of these pairs, we simply square the resulting value, which gives us formula_23 functionally distinct positions. 
There are formula_39 unreachable positions. 
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 formula_6 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 formula_34 unreachable positions and formula_70 reachable positions.
More than two players.
Given a roll-over of 5 and 2 hands.