Hi, all!

I am finishing proof of 11-clue valid puzzles non-existance.

There are 2 remaining map types (type 6 and type 7) only. It is easy to prove non-existance of such maps.

Let's consider type 6.

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`Type 6`

X 0 0

0 X X

X X X

Type 6 maps have the next clue distribution (map of type 6 always has 6 non-empty boxes) after applying composition rule 2 - every distribution must have at least 1 box containing not less than 4 clues.

A. 1 1 1 1 1 6 - 5 boxes contain 1 clue, 1 box contains 6 clues.

B. 1 1 1 1 2 5 - 4 boxes contain 1 clue, 1 box contains 2 clues, 1 box contains 5 clues.

C. 1 1 1 1 3 4 - 4 boxes contain 1 clue, 1 box contains 3 clues, 1 box contains 4 clues.

D. 1 1 1 2 2 4 - 3 boxes contain 1 clue, 2 boxes contain 2 clues, 1 box contains 4 clues.

Variants "A", "B", "C" have 1 or 2 boxes having greater than 1 clue only. Hence maps of these variants must have at least 1 band and 1 stack with all boxes having not greater than 1 clue. So, maps of these variants are subsets of Magic Pattern and are prohibited by composition rule 3.

Here are all possible maps for distribution "D".

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`Type 6`

N1 N2 N3 N4 N5 N6

4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0

0 1 1 0 2 1 0 2 2 0 1 1 0 2 1 0 1 2

2 2 1 2 1 1 1 1 1 1 2 2 1 2 1 1 2 1

Map N1 is prohibited by composition rule 3 (band B456 + stack B369). Map N2 is prohibited by composition rule 6 (band B456 + stack B369). Map N3 is prohibited by composition rule 6 (band B789 + stack B369). Maps N4, N5 are prohibited by composition rule 6 (band B456 + stack B369). Map N6 is prohibited by composition rule 7 (band B456 + stack B369). So, distribution "D" of type 6 has no valid puzzles.

Therefore, type 6 of 11-clue maps has no valid puzzles.

Let's consider type 7.

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`Type 7`

X 0 0

0 X X

0 X X

Type 7 maps have the next clue distribution (map of type 7 always has 5 non-empty boxes) after applying composition rule 2 - every distribution must have at least 1 box containing not less than 4 clues.

A. 1 1 1 1 7 - 4 boxes contain 1 clue, 1 box contains 7 clues.

B. 1 1 1 2 6 - 3 boxes contain 1 clue, 1 box contains 2 clues, 1 box contains 6 clues.

C. 1 1 1 3 5 - 3 boxes contain 1 clue, 1 box contains 3 clues, 1 box contains 5 clues.

D. 1 1 1 4 4 - 3 boxes contain 1 clue, 2 boxes contain 4 clues.

E. 1 1 2 2 5 - 2 boxes contain 1 clue, 2 boxes contain 2 clues, 1 box contains 5 clues.

F. 1 1 2 3 4 - 2 boxes contain 1 clue, 1 box contains 2 clues, 1 box contains 3 clues, 1 box contains 4 clue.

G. 1 2 2 2 4 - 1 box contains 1 clue, 3 boxes contain 2 clues, 1 box contains 4 clues.

Variants "A", "B", "C", "D" have 1 or 2 boxes having greater than 1 clue only. Hence maps of these variants must have at least 1 band and 1 stack with all boxes having not greater than 1 clue. So, maps of these variants are prohibited by composition rule 3.

Here are all possible maps for distribution "E".

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`Type 7`

N1 N2

5 0 0 5 0 0

0 1 1 0 2 1

0 2 2 0 1 2

Maps N1, N2 are prohibited by composition rule 4 (band B456 + stack B369).

Here are all possible maps for distribution "F".

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`Type 7`

N1 N2

4 0 0 4 0 0

0 1 1 0 2 1

0 2 3 0 1 3

Map N1 is prohibited by composition rule 4 (band B456 + stack B258). Map N2 is prohibited by composition rule 7 (band B456 + stack B258).

Here is the only possible map for distribution "G".

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`Type 7`

N1

4 0 0

0 1 2

0 2 2

Map N1 is prohibited by composition rule 4 (band B456 + stack B258).

Therefore, type 7 of 11-clue maps has no valid puzzles, and

11-clue valid puzzles don't exist.

Summary of composition rules.

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`Valid puzzles composition rules`

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

Rule 1. Map of any valid puzzle must be isomorphic to one of these maps:

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

Rule 2. Map posted below has no valid puzzles when A < 4

A 0 0

9 9 9

9 9 9

Rule 3. Map posted below has no valid puzzles

1 1 1

1 9 9

1 9 9

Rule 4. Map posted below has no valid puzzles

1 2 0

2 9 9

0 9 9

Rule 5. Map posted below has no valid puzzles

0 1 2

1 9 9

2 9 9

Rule 6. Map posted below has no valid puzzles

1 2 0

1 9 9

1 9 9

Rule 7. Map posted below has no valid puzzles

2 1 0

1 9 9

0 9 9

Serg