Consider the following two player game based on the Euclidean algorithm: Positions are given by $(a,b)$ in $\mathbb N^2\setminus\{(0,0)\}$ (where $\mathbb N=\{0,1,2,\ldots\}$) defining a greatest common divisor in $\mathbb N$. Moves are given by $(a,b)\longrightarrow (\min(a,b),\max(a,b)-k\min(a,b))$ for $k$ in $\{1,\ldots,\lfloor \max(a,b)/\min(a,b)\rfloor\}$ where we assume $\min(a,b)>0$. Two players move in turn until arriving at $(d,0)$ or $(0,d)$. At this point, the player who can no longer move announces the greatest common divisor $d$ of $a$ and $b$ defining the initial position $(a,b)$ and wins.
Winning positions are easy to describe: $(a,b)$ is winning if and only if $\min(a,b)/\max(a,b)$ is smaller than the inverse $(-1+\sqrt{5})/2$ of the golden number $(1+\sqrt{5})/2$.
Has this game be described somewhere?
The variation where the last player with a move (i.e. playing a position $(a,b)$ with $\min(a,b)$ a divisor of $\max(a,b)$) wins is also easy to describe: A position $(a,b)$ is winning if either $(a=b)$ or if $ab>0$ and $\max(a,b)/\min(a,b)>(1+\sqrt{5})/2$.
