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 Post subject: Cells Counting Double
PostPosted: Wed Dec 30, 2015 9:51 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
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Location: Lethbridge, Alberta, Canada
These posts were prompted by two recent Human Solvables posted by HATMAN.

However I'll start with a simple example first, Afmob's Assassin 122
Image

Note the ring of cages around the outside edges of the cage pattern. Their cage total = 170. Each of the edges totals 45, however each of the corner cells count toward both a row and a column -> R19C19 must total 180 - 170 = 10. And since this is a Killer-X, each of the corner cells "sees" the other three corner cells so the four corner cells must have different values -> R19C19 = {1234}. And since R19C19 also "see" R5C5 then there's no 1,2,3,4 in R5C5.

I think there may be a similar posted puzzle for which the outside edges total 150 -> R19C19 must total 180 - 150 = 30 -> R19C19 = {6789}.

Edge totals of 151 and 169 also give immediate combinations for R19C19 of {5789} and {1235}. For other edge totals between 152 and 168 there will still be a disjoint cage R19C19.


Note that a continuous ring of cages around the outside edge, without the puzzle being a Killer-X, may also lead to immediate results in a few cases. When the edge total is 146 the corner cells will total 34 so R1C19 = {89} and R9C19 = {89}. Similarly when the edge total is 174 the corner cells will total 6 so R1C19 = {12} and R9C19 = {12}.

My Outies posts in this Killer Techniques forum was an example of a technique which isn't programmed into SudokuSolver. The technique covered by the posts in this thread seems to be another one not programmed into SudokuSolver, and possibly other software solvers.


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PostPosted: Thu Dec 31, 2015 12:00 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
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Location: Lethbridge, Alberta, Canada
And now moving on to HATMAN's Human Solvables; for anyone not acquainted with these puzzles, they are ones which humans should be able to solve more easily than software solvers, either by spotting interesting interactions or because humans can use techniques which aren't programmed into software solvers.

Here is the position for Human Solvable 22 Sixes X 2 after step 3 of my walkthrough.

Image

In my walkthrough I made quite a lot of routine steps before spotting the important Human Solvable step. However it can be done immediately from this position.

Total around outer ring = 180 with corner cells counting double. Total of the four outer-edge 6-cell cages plus 1,2 pairs in R1 and C1 and the 3s in R9 and C9 = 164. This leaves 16 for the four corner cells, one of R56C9 and one of R9C45. R1C1 = 3 and min R1C9 + R9C1 = 9 -> max one of R56C9 + one of R9C45 + R9C9 = 4 -> one of R56C9 = 1, one of R9C45 = 1, R9C9 = 2, R1C9 + R9C1 = 9 = {45}, locked for D/

Thanks to Ed for providing the diagram. :)


For a harder example look at Human Solvable 23 Fives X
Hint:
This time it's not just the four corner cells which count double.
For more details see wellbeback's walkthrough or step 10 of my walkthrough.


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