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Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "sprouts without midpoints". One starts with $n$ points in the plane. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The lines are not allowed to intersect, i.e. the graph should be planar. The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "Sprouts without midpoints". One starts with $n$ points in the plane. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The lines are not allowed to intersect, i.e. the graph should be planar (otherwise the game would be too easy to solve). The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "non-planar sprouts without midpoints". One starts with $n$ points in space. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

The game can also be described via multisets of numbers $\leq 3$ (containing the point degrees), starting with $\{0,\dotsc,0\}$. Then it becomes a number-theoretic game. A move replaces some element $a$ by $a+2$ (if $a \leq 1$) or replaces two elements $a,b$ by $a+1,b+1$ (if $a,b \leq 2$). For $n=2$ the games are

$\begin{array}{c} \{0,0\} \to \{2,0\} \to \{3,1\} \to \{3,3\} \\\{0,0\} \to \{2,0\} \to \{2,2\} \to \{3,3\} \\ \{0,0\} \to \{1,1\} \to \{3,1\} \to \{3,3\} \\ \{0,0\} \to \{1,1\} \to \{2,2\} \to \{3,3\}\end{array}$

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "sprouts without midpoints". One starts with $n$ points in the plane. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The lines are not allowed to intersect, i.e. the graph should be planar. The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "non-planar sprouts without midpoints". One starts with $n$ points in space. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

The game can also be described via multisets of numbers $\leq 3$ (containing the point degrees), starting with $\{0,\dotsc,0\}$. Then it becomes a number-theoretic game. A move replaces some element $a$ by $a+2$ (if $a \leq 1$) or replacing two elements $a,b$ by $a+1,b+1$ (if $a,b \leq 2$). For $n=2$ the games are

$\begin{array}{c} \{0,0\} \to \{2,0\} \to \{3,1\} \to \{3,3\} \\\{0,0\} \to \{2,0\} \to \{2,2\} \to \{3,3\} \\ \{0,0\} \to \{1,1\} \to \{3,1\} \to \{3,3\} \\ \{0,0\} \to \{1,1\} \to \{2,2\} \to \{3,3\}\end{array}$

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "non-planar sprouts without midpoints". One starts with $n$ points in space. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

The game can also be described via multisets of numbers $\leq 3$ (containing the point degrees), starting with $\{0,\dotsc,0\}$. Then it becomes a number-theoretic game. A move replaces some element $a$ by $a+2$ (if $a \leq 1$) or replaces two elements $a,b$ by $a+1,b+1$ (if $a,b \leq 2$). For $n=2$ the games are

$\begin{array}{c} \{0,0\} \to \{2,0\} \to \{3,1\} \to \{3,3\} \\\{0,0\} \to \{2,0\} \to \{2,2\} \to \{3,3\} \\ \{0,0\} \to \{1,1\} \to \{3,1\} \to \{3,3\} \\ \{0,0\} \to \{1,1\} \to \{2,2\} \to \{3,3\}\end{array}$

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.