For a $9\times9$ grid $G$ (not necessarily respecting sudoku rules), a subset of cells $S$ is defined to be sudoku-friendly if the values in $S$ do not contradict the rules of sudoku.
This means that in $S$ no number occurs twice in the same row, column or box.
Take the grid $G$:
111111111 222222222 333333333 444444444 555555555 666666666 777777777 888888888 999999999
Then the following is an example of a sudoku-friendly subset $S$ of $G$:
1 2 3 4 5 6 7 8 9
A net is a mapping on a grid that produces a subset of cells. You can imagine it like a piece of paper with holes in it, lying on the grid, which hides some cells.
This is usually visualized with O (visible) and X (blocked) cells.
A net has a visible set, $V_N(G)$ which is defined as the set of cells in a grid $G$ which map to an O in $N$, namely the set of visible cells.
The net which retrieves the above visible set $S$ would look like this:
OXXXXXXXX XOXXXXXXX XXOXXXXXX XXXOXXXXX XXXXOXXXX XXXXXOXXX XXXXXXOXX XXXXXXXOX XXXXXXXXO
A sudoku net is a net whose visible set is sudoku-friendly.
Construct a sudoku net $N$ of maximum and minimum possible blocks such that there exists some 9x9 $G$ that is an impossible sudoku puzzle (namely no solution).
-Bounty awarding question:
- is there a grid which has unique solution whatever are the arrangements of legal patterns in friendly-sudoku ?