So for an arbitrary $n^2 \times n^2$ Sudoku puzzle, there are rules for inferring values and auxillary information in cells that run in polynomial time, e.g. using the filled in values to check to see if only one number is possible for a cell, using the filled in values to check if only one cell in a given row/column/square can contain a particular number, eliminating a number as a possibility for a row/column outside of a square if the number must lie in the row/column within the square, (and vice-versa for eliminating possibilities within a square), and taking advantage of the fact that if there are $k$ cells in a row/column/square that can only contain a particular subset of $k$ numbers, then no other cells in the row/column/square can contain those numbers. Presumably there are other polynomial time rules too.
The rule/strategy I want to exclude from this list is guessing a fixed number of cells' values, and checking to see if a contradiction is found or a complete valid puzzle is obtained. So, if we consider a certain reasonably rich set of polynomial time rules except for that one, what is the worst-case known recursion depth for how many cells' values need to be guessed simultaneously in order to make progress in a puzzle? Is this known for classical $n = 3$ for a reasonably rich certain set of polynomial time rules (e.g. the rules I listed above), and more interestingly for general $n$?
