= Nimber multiplication
{title2=$a\otimes b$}
= Nim-product
{c}
{synonym}
For finite <nimbers>, multiplication is recursively defined from smaller arguments by
$$
a\otimes b=\operatorname{mex}\{(a'\otimes b)\mathbin\oplus(a\otimes b')\mathbin\oplus(a'\otimes b'):a'<a,\ b'<b\},
$$
where $\oplus$ is <Nim-sum> and <mex> selects the smallest missing <nonnegative integer>. Thus zero times anything is zero and one times $b$ is $b$. For $a=b=2$, the displayed set is $\{0,2,3\}$, so $2\otimes2=1$.
Together with <Nim-sum>, this is field arithmetic on finite nimbers, rather than ordinary integer multiplication. In particular $2\otimes2=1$ cannot be computed by a usual integer product or bitwise <exclusive or>.https://circles.math.ucla.edu/circles/lib/data/Handout-4717-4806.pdf[Aaron Anderson, Nimbers, 3 October 2021, multiplication definition]{ref}
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