The SudokuP puzzles which have been occupying much of our recent discussion are just one particular instance of a more general class of puzzles in which a 4th dimension (psets) is added to the 3 dimensions (rows, columns, boxes) of standard Sudoku.

This 4th dimension generally can be any partiton of the 81 cells into 9 psets. SudokuP partitions the grid into 9 psets, where each pset contains 9 cells that are in the same box-position. The pset mapping in this case looks like this (where cells in the same pset are assigned the same letter):

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`a b c | a b c | a b c`

d e f | d e f | d e f

g h i | g h i | g h i

---------------------

a b c | a b c | a b c

d e f | d e f | d e f

g h i | g h i | g h i

---------------------

a b c | a b c | a b c

d e f | d e f | d e f

g h i | g h i | g h i

In SudokuPX, we extend this to arbitrary partitions of the grid into 9 disjoint sets.

One measure of interest is the number of cells whose domain is affected by a digit being assigned to any cell. Brian Taylor's BB solver calls these the associated cells. For standard Sudoku the associated cell count (ACC) is 20 (8 cells in the row, 8 in the column, and 6 in the box).

In SudokuP we have ACC = 24, since each cell has, in addition to the standard 20, affects 4 cells in the corresponding box-position, for those 4 boxes not in the same band/stack as the cell whose value is being set.

The maximum possible ACC value would seem to be 28, when each cell affects the domain of 8 cells in addition to the standard 20. This is only possible where the psets constitute a Sudoku square, ie each pset has one cell from each row, column and box.

These psets therefore form a Sudoku grid that is orthogonal to the original grid.

We might expect that puzzles with higher ACC values require less clues, and indeed this is the case. While we search for the existence (or non-existence) of 10-clue SudokuP puzzles, a search expected to weeks, if not months, to complete, it takes only a few minutes to identify 8-clue SudokuPX puzzles, and to find that there a whole swag of them. Here is just one example:

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`Puzzle Psets Solution`

=========================================================================

. . . | 9 . . | . . . a b c | e i f | g h d 3 6 8 | 9 7 5 | 4 1 2

. . . | . . . | 7 . . d e f | g a h | b c i 1 5 4 | 3 8 2 | 7 9 6

. . . | . 4 . | . . . g h i | b d c | e f a 7 9 2 | 1 4 6 | 8 3 5

--------------------- --------------------- ---------------------

. 1 . | . . . | . . . b f a | h e i | c d g 8 1 6 | 4 2 3 | 5 7 9

. . . | . . . | . . . e g d | c f a | h i b 4 2 5 | 7 9 1 | 6 8 3

. . . | 6 . . | . . . i c h | d b g | f a e 9 3 7 | 6 5 8 | 2 4 1

--------------------- --------------------- ---------------------

. . . | . . . | . . . h a e | f g d | i b c 5 7 3 | 8 6 9 | 1 2 4

2 . . | . . . | . . . c d g | i h b | a e f 2 8 1 | 5 3 4 | 9 6 7

. . . | . . . | . 5 8 f i b | a c e | d g h 6 4 9 | 2 1 7 | 3 5 8

A suggested graphical form for these puzzles (suggestions for improvement would be welcome):

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