Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have been reading the Winning Ways, the bible (?) of Combinatorial Game Theory.

I tried to calculate some games of the form {L|R}. But it is not easy to me.

For example, I don't know what {$\uparrow$,$\ast$ | $\downarrow$,$\ast$} is.

The game, say $G$, is fuzzy and $G+G=0$.

So I thought that the game might be $\ast$.

But $G+\ast$ is still fuzzy. Moreover $G+*n ~||~ 0$. Thus $G$ is not an impartial game.

I think $G$ can be simplified to the comibination of several symbols like $\ast$ or nimbers. But I have no idea.

Teach me, please.

share|cite|improve this question
up vote 4 down vote accepted

Games $H$ such that $H=\{G_L|-G_L\}$ form a closed set under addition, satisfy $H+H=0$, and must be either equal to $0$ or fuzzy with $0$. Some of these are nimbers: $*=\{0|0\}, *2=\{0,*|0,*\}$, etc. Those that are not nimbers and not zero seem to be typically written as $\pm(G_L)$, such as $\pm 1 = \{1|-1\}$, so your game is $$G=\{\uparrow, * \big\vert\downarrow,*\} = \pm(\uparrow,*).$$ Your game and the result of adding $*$ to it, $$G+* = \pm(0,\uparrow*),$$ are two of the simpler such games, with birthday equal to $3$. They satisfy $$ \downarrow*\text{}<G<\text{}\uparrow*\qquad\text{and}\qquad\downarrow\text{}< G+*<\quad\uparrow, $$ giving an idea of how "unfuzzy" they are; but $G$ is fuzzy with $\uparrow$ and $\downarrow$. (Note that $*$ behaves exactly the same as $G$ in all these particulars $-$ it is fuzzy with $\uparrow$ and $\downarrow$, strictly between $\uparrow*$ and $\downarrow*$, and becomes between $\uparrow$ and $\downarrow$ after adding $*$ to it.)

share|cite|improve this answer
Thank you for your helful comment. – P.-S. Park Dec 20 '11 at 11:51

You can check that $G$ has no dominated or reversible options (this is easy once you've checked that $\uparrow*$, $G$, and $G + *$ are all fuzzy), so the form you've given is already the canonical form. Being in canonical form doesn't mean it couldn't be a sum of simpler objects, though.

However, in this case, $G$ doesn't simplify to any sum of ups, downs, or nimbers. Two (or more) ups plus a nimber is positive, whereas $G$ is not. A single up is also positive. The game $\uparrow *n$, if $n>1$, is also positive. The game $\uparrow *$ is fuzzy, but 2 of them added together give $\uparrow \uparrow$ which is positive whereas $G+G=0$.

share|cite|improve this answer
I see. Thank you, Ted. Is there any symbol for this game $G$? – P.-S. Park Dec 19 '11 at 7:31
I don't know of any symbol for it. The software CGSuite for studying combinatorial games, doesn't simplify it any further. – Ted Dec 20 '11 at 3:33
Thank you for your information about CGSuite. – P.-S. Park Dec 20 '11 at 11:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.