This relates to the answer to a question "Who wins two player sudoku?" and this awesome blog.

A Sudoku can be $N \times N$ where $\sqrt{N}$ is a natural number because $N \times N / \sqrt{N} \times \sqrt{N} = N$, the number of regions in the Sudoku.

It seems more computationally efficient to describe it as $(m^2)^2 = m^2\;(m^2) = m \times m\: (m \times m)$ because the square root does not have to be derived, and more notationally efficient because the regions are described.

$m \times m\; (m \times m)$ means "an $m$ by $m$ matrix of $m$ by $m$ matrices". $m^2\;(m^2)$ means "$m$ squared $m$ squared", a colloquial description of the geometry.

Coordinates (cells) are $\left[-\frac{m}{2},\ldots, \frac{m}{2}\right]$ for even $m$ and $\left[-\frac{m-1}{2},\ldots, \frac{m-1}{2}\right]$ for odd $m$.

Using these methods, I understand the boundaries between regions in any finite Sudoku.

• What I don't understand is the common boundaries between regions in an infinite Sudoku.

_{I find it helpful to think of the surface of the Sudoku as a torus, so the left side connects to the right side, and the top to the bottom, which does not violate the constraints but makes every regional boundary common.}

I'm also interested in how to describe an infinite Sudoku using cardinal numbers.

Using the above notation, $\infty = \aleph_0$ is the set of all natural numbers, I'd do it:

$$ (\infty^2)^2 = \infty\times\infty\;(\infty\times\infty) $$

I feel like $\sqrt{\infty}$ is problematic because it seems like $\sqrt{\aleph_0} = \aleph_0$

It also strikes me while the regions in infinite Sudoku can be $\aleph_0$, a Sudoku comprised of an such infinite regions is probably uncountable.

Since I am only confident in my understanding of $\aleph_0$, spending most of my time on finite sets, is the Sudoku comprised of an infinite set of regions with an infinite number of cells an $\Omega$?

There's also the question of $(\infty^n)^n$ where $n = \aleph_0$