# Does there exist a non-zero game such that the sum of three or more copies of it is zero?

In combinatorial game theory, there are non-zero games $G$ with the property $G+G=0$; this is in particular true for all impartial games.

Now I wonder if there also exist non-zero games such that $G+G+G=0$. I don't see an obvious reason why those shouldn't exist, but I also have no idea on how to construct such a game.

Such a game obviously has to be a fuzzy game, as the sum of two positive games is positive, and the sum of two negative games is negative. Also, clearly $G\ne -G$, or else $G+G+G = G+(-G)+G = G \ne 0$. But beyond that, I'm out of ideas.

It would also be interesting for larger numbers than three. If for $n\in\mathbb N$ we define $nG$ as the sum of $n$ copies of $G$, for which (if any) $n>2$ does there exist a game such that $nG=0$, but $mG\ne 0$ for $0<m<n$?

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