Hi, all!

I continue proving non-existance of 11-clue valid puzzles.

Now I am considering type 5.

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`Type 5`

X X 0

X 0 X

0 X X

Type 5 maps have next clue by box distributions (map of type 5 always has 6 non-empty boxes).

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 clue.

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

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

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

G. 1 2 2 2 2 2 - 1 box contains 1 clue, 5 boxes contain 2 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.

To limit maps variants which must be checked out, I'll prove the statement that contents of any non-empty box of type 5 map can be moved to any other non-empty box by some validity preserving transformations.

Let's consider 2 VPT.

VPT 1

1. Swap stacks B147/B258.

2. Swap bands B456/B789.

VPT 2

1. Swap stacks B258/B369.

2. Swap bands B123/B456.

Here are illustrations for these VPTs.

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`VPT 1`

swap swap Box contents exchanges

A B 0 stacks B A 0 bands B A 0 B1 <---> B2 B4 <---> B8 B6 <---> B9

C 0 D ------> 0 C D -----> E 0 F

0 E F E 0 F 0 C D

VPT 2

swap swap Box contents exchanges

A B 0 stacks A 0 B bands C D 0 B1 <---> B4 B2 <---> B6 B8 <---> B9

C 0 D ------> C D 0 -----> A 0 B

0 E F 0 F E 0 F E

If we'll apply VPT 1 and VPT 2 in turn to type 5 map (VPT 1, VPT 2, VPT 1, ...), the contents of box B1 will move to other boxes in such sequence: B1 --> B2 --> B6 --> B9 --> B8 --> B4 --> B1. All non-empty boxes of type 5 map are represented in this sequence. So, there must exist a VPT sequence (VPT 1, VPT 2, ... , VPT N; where N - "1" or "2"), moving any non-empty box contents to any "target" non-empty box.

Let's analyze distribution "D".

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

We can move 4-clue box contents to any type 5 non-empty box. So, we can treat box B9 containing 4 clues.

Let's consider left top "corner" of type 5 map marked by letters "C":

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`C C 0`

C 0 X

0 X X

To be not excluded by available composition rules, such "corner" must contain 2 or 3 boxes having 2 clues. Only such maps are possible (up to isomorph.):

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` Type 5`

N1 N2

2 2 0 1 2 0

1 0 1 2 0 1

0 1 4 0 1 4

One can see that map N1, N2 are prohibited by composition rule 5 (band B456 + stack B258). So, distribution "D" of type 5 has no valid puzzles.

Let's analyze distribution "E".

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

Each of both "corners" of this distribution maps must contain the box having 3 clues, otherwise (one corner contains both boxes having 3 clues) the corner not containing boxes having 3 clues would contain 1 box having 2 clues only and several boxes having 1 clue. It is impossible (see beginning of this post). We can treat box B9 containing 3 clues. Only such maps are possible (up to isomorph.):

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` Type 5`

N1 N2 N3 N4 N5

3 2 0 2 3 0 1 3 0 1 3 0 1 3 0

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

0 1 3 0 1 3 0 1 3 0 1 3 0 2 3

Map N1 is prohibited by composition rule 5 (band B456 + stack B258), map N2 is prohibited by composition rule 4 (band B456 + stack B147), map N3 is prohibited by composition rule 7 (band B456 + stack B147), map N4 is prohibited by composition rule 4 (band B456 + stack B147), map N5 is prohibited by composition rule 3 (band B456 + stack B147). So, distribution "E" of type 5 has no valid puzzles.

Let's analyze distribution "F".

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

We can treat box B9 containing 3 clues. Left top corner must contain not less than 2 boxes having 2 clues, and cannot contain 1 clue in the box B1 (excluded by composition rule 4). Only such maps are possible (up to isomorph.):

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` Type 5`

N1 N2 N3

2 2 0 2 2 0 2 2 0

2 0 1 1 0 2 1 0 1

0 1 3 0 1 3 0 2 3

Maps N1, N2 are prohibited by composition rule 5 (band B456 + stack B258), map N3 is prohibited by composition rule 4 (band B456 + stack B147). So, distribution "F" of type 5 has no valid puzzles.

Let's analyze distribution "G".

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

We can move 1-clue box contents (by series of VPT 1, VPT 2, VPT 1, ... transformations) to B1 box, and we can see only one possible map:

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`1 2 0`

2 0 2

0 2 2

You can see that this map is prohibited by composition rule 4. So, distribution "G" of type 5 has no valid puzzles.

Therefore, type 5 of 11-clue maps has no valid puzzles.

(I hardly believe that anyone can check my statements. For what puposes am I writing my huge posts? I don't know.)

(Continuation follows)

Serg

[Edited: I corrected the error in my proof - distribution "G" for type 5 was missed. Thanks to

ronk for his correction]