I hope it works out !
If it helps my furthur thoughts are.....
Its all to do with the unavoidable sets - in an attempt to complete a band
- Code: Select all
+---+---+---+
|123|476|895|
|456|893|271|
|..9|152|346|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
|.7.|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
- Code: Select all
+---+---+---+
|...|...|...|
|...|...|...|
|78.|...|...|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
|87.|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
The 7 clue in box 7 is necessary to be a clue in this unavoidable set
- Code: Select all
+---+---+---+
|*..|...|...|
|...|..1|...|
|...|...|..1|
+---+---+---+
|...|...|...|
|.1.|...|...|
|...|...|...|
+---+---+---+
|...|...|...|
|...|...|...|
|..1|...|...|
+---+---+---+ thie clue * is inserted because of the 4 given clues
- Code: Select all
+---+---+---+
|12.|...|...|
|4*.|...|...|
|...|...|..5|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
|...|...|...|
|...|...|...|
|..5|...|...|
+---+---+---+ the clue * is inserted bcause of the 5 given clues
The given clues cover every unavoidable set that our inserted clue is in.
It is quite amazing that 17 clues can define a grid [all unavoidables have a clue in them]
With respect to completing a band.......
With 18 clues we only had 9 extra to define - this is easily possible with clues within the band.
With 17 clues we had 9 plus 1 more within a band to define - and we couldnt do it. We had to resort to inserting clues ouside the band, and use cross constraints to complete our band/grid.
- Code: Select all
+---+---+---+ +---+---+---+
|7.3|8..|...| |753|842|6..|
|...|6..|4..| |891|657|432|
|2..|...|...| |264|319|7..|
+---+---+---+ +---+---+---+
|...|...|.73| |186|294|573|
|.4.|1..|...| |547|1..|2..|
|...|...|...| |932|765|8..|
+---+---+---+ +---+---+---+
|61.|...|9..| |615|4..|927|
|...|.2.|1..| |379|526|1..|
|..8|.7.|...| |428|971|3..|
+---+---+---+ +---+---+---+ 17 clues solves the first vertical band
With 16 clues the gulf is "exponentially" wider. We know that we are going to have to insert clues outside the band.
There will be clues in the band
There will be clues outside the band
We have two scenarios - comparing the clue distribution in our 16 attempt with our valid 17.
1.Number of clues within the band the same - but one less clue available to insert outside the band to cover the unavoidable sets.
2.One less clue in the band - one extra clue in the band which has to be covered by one less clue within the band and a similar number of clues outside the band.
If it is difficult as I think it is to cover the band with 17 clues - doing it with 16 is going to be impossible.
If we can never fill a band with 16 clues.......we can never complete a grid !
When we can get rid of the "if" it will be proof !
C