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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 $n = \aleph_0$

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 $n = \aleph_0$

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

A Sudoku can be N * N where √N is a natural number because N * N / √N * √N = N, the number of regions in the Sudoku.

It seems more computationally efficient to describe it as (m²)² = m²(m²) = m * m (m * m) because the square root does not have to be derived, and more notationally efficient because the regions are described.

m * m (m * m) means "an m by m matrix of m by m matrices". m²(m²) means "m squared m squared", a colloquial description of the geometry.

Coordinates (cells) are [-ᵐ∕₂, .., ᵐ∕₂] for even m and [-ᵐ⁻¹∕₂, .., 0, .., ᵐ⁻¹∕₂] 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, ∞ = אₒ is the set of all natural numbers, I'd do it:

(∞²)² = ∞ * ∞ (∞ * ∞)

I feel like √∞ is problematic because it seems like √אₒ = אₒ

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

Since I am only confident in my understanding of אₒ, 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 Ω?

There's also the question of (∞ⁿ)ⁿ where n = אₒ

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 $n = \aleph_0$