Post History
#2: Post edited
by
trichoplax
·
2026-05-06T12:28:57Z (4 months ago)
Suggest a Markdown footnote that automatically links in both directions
Are there combinatorial games $G$ such that $G+G$ is fuzzy?
- In giving an incomplete answer to [this question](https://math.codidact.com/posts/287192), on the existence of nonzero combinatorial games $G$ such that $nG=0$ and $mG≠0$ for $m < n$, I noted a hypothesis that would immediately solve the problem (in the negative):
- \[G+G\text{ is non-fuzzy for all }G.\]
<sup>1</sup>- I've verified this hypothesis for all the combinatorial games of depth 2. It is always true for symmetric games (in which $G+G = 0$). It is also true whenever Left's best move in the game $G$ is better for Left than Right's best move is for Right (because then $G+G$ is a win for Left), and *mutatis mutandis* when we switch Right and Left.
- Is it true, though? Can we prove it or give a counterexample for it?
- ---
<sup>1</sup> The reason this hypothesis would solve the [original problem](https://math.codidact.com/posts/287192) is as follows. If $G+G$ is always non-fuzzy, then $nG=0$ *only* if $G+G=0$; otherwise $G+G$ would have to be either positive or negative, and in either case $nG$ could not equal 0.
- In giving an incomplete answer to [this question](https://math.codidact.com/posts/287192), on the existence of nonzero combinatorial games $G$ such that $nG=0$ and $mG≠0$ for $m < n$, I noted a hypothesis that would immediately solve the problem (in the negative):
- \[G+G\text{ is non-fuzzy for all }G.\]
- [^1]
- I've verified this hypothesis for all the combinatorial games of depth 2. It is always true for symmetric games (in which $G+G = 0$). It is also true whenever Left's best move in the game $G$ is better for Left than Right's best move is for Right (because then $G+G$ is a win for Left), and *mutatis mutandis* when we switch Right and Left.
- Is it true, though? Can we prove it or give a counterexample for it?
- ---
- [^1]: The reason this hypothesis would solve the [original problem](https://math.codidact.com/posts/287192) is as follows. If $G+G$ is always non-fuzzy, then $nG=0$ *only* if $G+G=0$; otherwise $G+G$ would have to be either positive or negative, and in either case $nG$ could not equal 0.
#1: Initial revision
by
clemens
·
2026-05-06T03:00:54Z (4 months ago)
Are there combinatorial games $G$ such that $G+G$ is fuzzy?
In giving an incomplete answer to [this question](https://math.codidact.com/posts/287192), on the existence of nonzero combinatorial games $G$ such that $nG=0$ and $mG≠0$ for $m < n$, I noted a hypothesis that would immediately solve the problem (in the negative):
\[G+G\text{ is non-fuzzy for all }G.\]
<sup>1</sup>
I've verified this hypothesis for all the combinatorial games of depth 2. It is always true for symmetric games (in which $G+G = 0$). It is also true whenever Left's best move in the game $G$ is better for Left than Right's best move is for Right (because then $G+G$ is a win for Left), and *mutatis mutandis* when we switch Right and Left.
Is it true, though? Can we prove it or give a counterexample for it?
---
<sup>1</sup> The reason this hypothesis would solve the [original problem](https://math.codidact.com/posts/287192) is as follows. If $G+G$ is always non-fuzzy, then $nG=0$ *only* if $G+G=0$; otherwise $G+G$ would have to be either positive or negative, and in either case $nG$ could not equal 0.