coloin wrote:By definition all the 416 bands are compatible with sudokuP .

That is true, and each of them has (many) extensions to a complete grid.

With SudokuP, though, there are 6585 "minlex canonical" bands ... or "band types".

The larger number, is due to there only being 2*6^3 transformations (as opposed to 6^5), that can be used in canonicalization.

The factor of 2, comes from the "F" transformation ... mapping bands to bands.

For the uninitiated, F swaps mini-rows 2 & 4, 3 & 7, and 6 & 8 ... separately, in each band, when it's applied to a full grid (or puzzle).

The canonical grid catalog:

When the minlex canonical grids are listed in minlex order, the list splits into sections with the same (minlex) canonical band, in the band 1 position.

As it was with standard Sudoku, the "would be" sections for some of the 6585 canonical/minlex band types, are "empty".

Those bands don't have any extension to a minlex canonical grid.

The number of non-empty sections, is 5067, and the first empty section, would be for canonical band index #3312.

Again, as it was with standard Sudoku, near the beginning of the list, the "same band 1" sections are large ... and near the end, they're small.

The first section, contains 1,268,254 grids. It completely covers Mathimagics' first (1M grid) "interval", from above. The 2nd section contains 3,440,695. The last interval that Mathimagics described ... grids 53000001-53666690 ... contains 4012 complete, non-empty, "same band 1" sections, and part of another.

'MinClues' values for SudokuP bands:

In 2011 -- is it really that long ago ! -- Mladen published a table of information about (standard) Sudoku bands -- Bands and low-clue puzzles.

Of particular interest here, is the "MinClues" column.

It shows the (genuine) minimum number of clues that are needed to hit every unavoidable set in the band.

(The MinClues number is >= the "MCN" number for the "in band" UA sets).

For a puzzle with a given grid as its solution, each band must contain a number of clues that's >= the MinClues value for its "band type".

Note: As was definitely not the case with standard Sudoku, many of the bands -- 1718 of them -- have "MinClues=0".

(For standard Sudoku, the values were always >= 2).

There's a (zipped) file attached below.

The file lists, for each band type (in minlex order):

- The 27 cell values for the band.

The number of canonical grids having the band in the band 1 position.

A running subtotal of the above -- from the subtotal, you can tell where the "same band" sections fall, in Mathimagics' "interval" scheme.

The "MinClues" value for the band.