Hi, all!

It looks like there is no proof of non-existance of 8-clue minimal valid puzzles. Ok, I'd like to propose my own approach based on "puzzle composition rules".

First, I have to define new term "puzzle map".

Let's consider a puzzle and compute numbers of clues for every box of sudoku grid. We'll obtain 9 digits (minimal possible value is 0, maximal possible value is 9). If we'll arrange these digits in 3 x 3 matrix, in correspondence with boxes positions, we'll get

puzzle map. For example, let's consider (invalid) 16-clue puzzle

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` . . . | . . . | 9 6 .`

6 9 . | . 1 . | 8 . .

. . 2 | . . . | 1 . .

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

4 . . | . . 9 | . 8 .

. . 9 | . . . | . . .

. . . | . . . | . 1 .

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

. . . | . . . | . . .

. 4 . | . . . | . . .

. . . | . 4 . | . . 9

This puzzle has such puzzle map:

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`3 1 4`

2 1 2

1 1 1

Variable (unknown), but non-zero number of clues in a box can be denoted by letter. For example, puzzle map (or simply "map")

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`X 0 0`

X X X

X X X

denotes a class of puzzles having non-zero numbers of clues in the boxes B1, B4-B9 and having no clues in the boxes B2 and B3. This notation was not so new (see well-known thread

Empty Boxes - I).

Let's classify all possible

valid puzzles by number of empty boxes (taking in account their arrangements to get nonisomorphic cases). There are 7 nonisomorphic types of valid puzzles only:

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`Type 1 Type 2 Type 3 Type 4 Type 5 Type 6 Type 7`

X X X X X 0 X 0 0 X X 0 X X 0 X 0 0 X 0 0

X X X X X X X X X X X X X 0 X 0 X X 0 X X

X X X X X X X X X 0 X X 0 X X X X X 0 X X

(I've copied

Ocean's list from thread

Empty Boxes - I, adding obvious "Type 1" and rearranging empty boxes to get more convinient for further analysis configurations.)

Let's now consider 8-clue valid puzzles and account for each type all possible maps. Type 1 is impossible because 8-clue puzzle must contain at least 1 empty box. Type 3 is impossible too, because such patterns must contain at least 4 clues in the box B1 to have valid puzzles (see thread

Investigation of one-band-free patterns). Type 6 is impossible for the same reason as type 3.

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`Type 2`

1 1 0

1 1 1

1 1 1

Type 4

1 1 0 1 1 0 1 1 0

1 1 1 1 1 1 1 2 1

0 1 2 0 2 1 0 1 1

Type 5

1 1 0 1 1 0

1 0 1 1 0 2

0 2 2 0 2 1

Type 7

4 0 0

0 1 1

0 1 1

Now we should prove impossibility of 8-clue valid puzzles having one of the possible maps posted above.

I hope I was not wrong in determining of all possible maps.

(Continuation follows.)

Serg