Yes, no, yes, no, yes, no, ...

I guess i changed my mind 10 times, how this question should be answered:

Is it possible to enumerate all non equivalent puzzles from a set of all non equivalent sub-puzzles ?

My answer now is

no again.

My last thought had been, that if you canonicalize sub-puzzles and complete puzzles to the same order of cells, i.e. find the minimal minlex form for a predefined order (where all sub-puzzle cells come before the rest), then for each complete puzzle there is a morph, for which you can find a numbering of the sub-puzzle cells, which is minimal for the sub-puzzle (just minlex these cells first and then renumber the rest accordingly).

But this is not true, when sub-puzzle cells are in the same unit as complementary cells.

Example:

I defined the cell order as

32,40,48,50,2,6,74,78,18,26,54,62,14,34,46,66,10,16,64,70

For this order this 16-clue sub-puzzle has that minimal minlex form:

- Code: Select all
`..2...1.......2...4.......6.....1.3.....2.....2.3.4...7.......8...2.......3...5..`

..2...1.......2...4.......6.....1.2.....2.....2.3.4...7.......8...1.......3...5..

But this puzzle

- Code: Select all
`..2...1...1...2.4.4.......6.....1.3.....2.....2.3.4...7.......8.4.2...1...3...5..`

has as minimal minlex of its sub-puzzle the first one. Its sub-puzzle never can be morphed/renumbered to a minimal one. So if you throw away this first sub-puzzle, you have no chance to find the complete puzzle or an equivalent by adding numbers to any non equivalent sub-puzzles.

- Code: Select all
`+-------+-------+-------+ +-------+-------+-------+ +-------+-------+-------+`

| . . 5 | . . . | 6 . . | | . . 2 | . . . | 1 . . | | . . 2 | . . . | 1 . . |

| .17 . | . .13 | .18 . | | . 1 . | . . 2 | . 4 . | | . . . | . . 2 | . . . |

| 9 . . | . . . | . .10 | | 4 . . | . . . | . . 6 | | 4 . . | . . . | . . 6 |

+-------+-------+-------+ +-------+-------+-------+ +-------+-------+-------+

| . . . | . . 1 | .14 . | | . . . | . . 1 | .*3 . | | . . . | . . 1 | .*2 . |

| . . . | . 2 . | . . . | | . . . | . 2 . | . . . | | . . . | . 2 . | . . . |

| .16 . | 3 . 4 | . . . | | . 2 . | 3 . 4 | . . . | | . 2 . | 3 . 4 | . . . |

+-------+-------+-------+ +-------+-------+-------+ +-------+-------+-------+

|11 . . | . . . | . .12 | | 7 . . | . . . | . . 8 | | 7 . . | . . . | . . 8 |

| .19 . |15 . . | .20 . | | . 4 . |*2 . . | . 1 . | | . . . |*1 . . | . . . |

| . . 7 | . . . | 8 . . | | . . 3 | . . . | 5 . . | | . . 3 | . . . | 5 . . |

+-------+-------+-------+ +-------+-------+-------+ +-------+-------+-------+

cell order puzzle with min. sub-puzzle normalized sub-puzzle