I was recently reminded of the jigsaw sudoku, which is a sudoku variant where each of the nine regions that need to be filled with the numbers from 1 to 9 are irregular regions instead of square regions. This made me wonder about how these puzzles are constructed, and whether there will always be a way to construct a jigsaw sudoku given a filled 9-by-9 grid where each of the numbers 1 to 9 appear exactly once on each row and each column in the grid.
More generally, let $M_n$ be a Latin square of size $n$ (i.e. $M_n$ contains every natural number from $1$ up to $n$ as an entry in each row and each column).
A collection of $k$ connected (by travelling horizontally and/or vertically, but not diagonally) entries in $M_n$ is called a $k$-region. If $R$ is an $n$-region formed from a single row or column in $M_n$, $R$ is called trivial. If $n$ is a perfect square and $R$ is an $n$-region formed from a square submatrix of $M_n$, $R$ is called regular. If an $n$-region $R$ is neither trival or regular, $R$ is called irregular.
Suppose $\Sigma_n$ is a collection of $n$ non-overlapping $n$-regions $R_i$ formed from $M_n$, such that each $R_i\in \Sigma_n$ is formed from entries with values ranging from $1$ up to $n$. Then the pair $(M_n,\Sigma_n)$ is called an $n$-sudoku template. If each $R_i\in\Sigma_n$ is trivial, then $(M_n,\Sigma_n)$ is said to be trivial; if $n$ is a perfect square and each $R_i\in\Sigma_n$ is regular, then $(M_n,\Sigma_n)$ is said to be regular. Otherwise, if each $R_i\in\Sigma_n$ is irregular, then $(M_n,\Sigma_n)$ is said to be chaotic.
It quickly becomes apparent that there cannot be a chaotic $n$-sudoku template with $n\leq 3$ but I have a hard time figuring out what happens for larger values of $n$.
Also immediately apparent is that no matter what, if $M_n$ is a Latin square there is always a $\Sigma_n$ such that $(M_n,\Sigma_n)$ is a trivial $n$-sudoku template.
Q1. Suppose that $n\geq 3$ and that $M_n$ is a Latin square of size $n$. Is it always possible to find a $\Sigma_n$ such that the $n$-sudoku template $(M_n,\Sigma_n)$ is not trivial? If not, is there any number $n$ enabling us to find such a $\Sigma_n$ no matter how $M_n$ looks?
If the answer to the above question is positive, then the following question may be relevant:
Q2. Suppose that $n\geq 4$ and that $M_n$ is a Latin square of size $n$. Is it always possible to find a $\Sigma_n$ such that the $n$-sudoku template $(M_n,\Sigma_n)$ is chaotic? If not, is there any number $n$ enabling us to find such a $\Sigma_n$ no matter how $M_n$ looks?
I have no idea how to answer the above questions, but I suspect that maybe graph/Ramsey theory could be required at some point.