The New Sudoku Players' Forum

[999_Springs]

What is the maximum number of clues for puzzles of a given ER rating?

I'll get this thread started...

Code: Select all

rating |clues
-------+-----
 1.0   | 80
 1.2   | 77
 .
 .
 .

[RW]

Not easy to find out for most ratings (and should probably be maximum known number of clues), but I'll continue the list anyway:

Code: Select all

rating |clues
-------+-----
 1.0   | 80
 1.2   | 77
 .
 .
 .
 11.0  | 37
 600002059520040010003500200300194500010658030005273001004005100030020045750400008

And if I recall correctly, the maximum number of clues for anything above 1.5 should be 72.

RW

[JPF]

Here is a quick contribution, probably easy to improve:

Code: Select all

7.2 # 40 # 170000465905060802640000390780000100091683054000200083004000531507008640300005078
6.6 # 46 # 180030470030804012054010003290063150501298346403100298900581004840300060300040800
5.6 # 46 # 102637005609008001074195026005001703030070054760350210020006500516083942803000067
4.5 # 40 # 080009405007080610609107038000008950000060740900010002760800304538400107204070586
4.2 # 45 # 190630000756081309430295706007906801560018900819000607000803194900100560000009270


And if I recall correctly, the maximum number of clues for anything above 1.5 should be 72.

More precisely, the conjecture is :
if a valid puzzle has more than 70 clues, then the puzzle is all singles.

JPF

[Glyn]

I would imagine that the maximum number of clues should decrease monotonically with an increase in the rating of a perfect assessing program. Any departure we see at the early stages of an investigation would be a manifestation of not having access to the optimum solving path and not evaluating the complete score for that path.

[JPF]

hmm...
You should have a look at this discussion.

JPF

[Glyn]

Thanks JPF I've seen puzzles get harder depending on the next move made deflecting the route, from what you're indicating extra clues can have the same effect on the route as well.

I would expect the hardest possible grid with N-1 givens to be harder than the hardest possible grid with N givens when assessed over all moves.

Obviously we can't test all starting grids given its vastness compared even to the number of solution grids. We also can't test all possible solving paths (even for one grid ). That's why I mentioned a perfect assessing program, which we will never have.

This looks like a good workout for SE.

[999_Springs]

if a valid puzzle has more than 70 clues, then the puzzle is all singles.

What's the ER rating of the 70-clue puzzle?

[JPF]

Here is an example given by gsf in an other forum
70 clues

Code: Select all

 1 2 3 | 4 5 6 | 7 8 9
 4 5 6 | 7 8 9 | 1 2 3
 7 8 9 | 1 3 2 | 5 6 4
-------+-------+-------
 2 3 . | 9 . 5 | 8 . 6
 6 . 5 | 2 . 8 | 9 3 .
 8 9 7 | 3 6 1 | 2 4 5
-------+-------+-------
 3 6 2 | 5 1 7 | 4 9 8
 5 . 8 | 6 9 4 | 3 . 2
 9 . . | 8 2 3 | 6 5 .



In this case, ER=5.6 or 6.6 (if you don't like the BUG)

JPF

[wintder]

Code: Select all

4 6 .|3 2 .|. 5 .
. . 5|6 7 8|. 4 3
7 . 3|4 5 .|6 . 2
-----+-----+-----
5 . 6|8 3 4|. 7 1
8 7 1|5 9 2|4 3 6
3 4 .|7 1 6|. . 5
-----+-----+-----
9 . 7|2 6 3|5 . 4
6 5 4|1 8 7|3 2 9
. 3 .|9 4 5|. 6 .  SE 3.2,  60 givens.

Code: Select all

6 5 3|8 2 4|1 9 7
9 1 8|7 3 6|2 4 5
2 4 7|. 5 .|6 8 3
-----+-----+-----
3 9 1|5 4 2|7 6 8
5 8 6|3 . .|4 2 1
4 7 2|6 . .|5 3 9
-----+-----+-----
. 2 4|. . 5|3 7 6
. 6 9|2 . 3|8 5 4
. 3 5|4 6 .|9 1 2   SE 4.2,  68 givens.

Code: Select all

9 1 4|3 2 6|5 8 7
. 6 .|8 7 4|3 1 9
. . .|1 9 5|4 6 2
-----+-----+-----
. . 1|6 3 2|9 4 8
8 4 6|9 5 1|2 7 3
. . 9|4 8 7|1 5 6
-----+-----+-----
1 . .|7 4 9|6 3 5
4 9 7|5 6 3|8 2 1
6 . .|2 1 8|7 9 4   SE  5.6,  68 givens.


6 . .|8 4 1|. 3 7
7 3 .|9 2 6|. 1 4
. . 4|7 3 5|. 6 2
-----+-----+-----
3 9 7|5 1 2|4 8 6
4 6 .|3 9 8|7 . 1
. . .|6 7 4|3 . 9
-----+-----+-----
. . .|4 8 3|6 7 5
8 4 6|2 5 7|1 9 3
5 7 3|1 6 9|2 4 8   SE 6.6, 64 givens.



. . 5|4 9 7|. 3 1
. . 9|2 3 1|. 7 .
3 7 1|5 8 6|4 9 2
-----+-----+-----
5 . .|. 6 8|1 2 7
. 1 7|. 4 .|. 6 .
8 6 .|7 1 .|9 4 .
-----+-----+-----
1 3 8|6 7 4|2 5 9
7 5 6|8 2 9|3 1 4
. . .|1 5 3|7 8 6    SE 5.8,  60 givens.


8 . 3|6 . .|7 . .
. 6 .|7 3 .|8 . .
7 . .|8 . .|6 3 1
-----+-----+-----
5 7 6|1 2 9|3 4 8
. . .|3 5 8|2 7 6
2 3 8|4 7 6|5 1 9
-----+-----+-----
3 2 1|5 6 4|9 8 7
6 8 5|9 1 7|4 2 3
9 4 7|2 8 3|1 6 5  SE 7.1,  64 clues.

1 . .|. 3 .|8 . 2
2 . 9|. 8 .|. . 6
8 5 .|. 2 .|. . 4
-----+-----+-----
3 7 2|5 9 4|6 1 8
4 9 5|6 1 8|2 7 3
6 8 1|3 7 2|5 4 9
-----+-----+-----
9 2 4|. 6 .|3 8 5
7 . .|8 5 9|4 2 1
5 1 8|2 4 3|9 6 7   SE 7.2,  62 givens.


. . 6|. . .|. . 1
2 . 3|1 . .|. 5 .
. . 1|. 3 .|. . .
-----+-----+-----
4 6 8|7 2 3|9 1 5
1 3 2|5 9 8|4 7 6
7 5 9|6 4 1|8 2 3
-----+-----+-----
9 1 4|3 6 2|5 8 7
3 2 7|8 5 9|1 6 4
6 8 5|4 1 7|. . .   SE 7.3,  59 givens.

[999_Springs]

Code: Select all

600002059520040010003500200300194500010658030005273001004005100030020045750400008

I put this puzzle into Bob Hanson's Sudoku Assistant/Solver. Without eliminating a single candidate it got stuck. Bob Hanson claims that he has never found a puzzle which his solver can't solve. Is there a typo in the puzzle or has Bob Hanson's solver been overwhelmed?

[ab]

Code: Select all

600002059520040010003500200300194500010658030005273001004005100030020045750400008

I put this puzzle into Bob Hanson's Sudoku Assistant/Solver. Without eliminating a single candidate it got stuck. Bob Hanson claims that he has never found a puzzle which his solver can't solve. Is there a typo in the puzzle or has Bob Hanson's solver been overwhelmed?

That puzzle definately has a unique solution. I put it into sudoku explainer and asked for the first step and it said it may take some time to work it out

[Glyn]

The 37 clue puzzle is ER11.0 for the first move. The puzzles that Bob Hanson quotes as solvable with Sudoku Assistant and Depth++ are simpler suggesting a greater depth is required for Bob's program in this case, his tests are quite old now.

[999_Springs]

I put it into sudoku explainer and asked for the first step and it said it may take some time to work it out

Bob Hanson's solver did that too after 22 seconds, but after 34 seconds the "solver is stuck" message appeared.

To see how his solver compares with SE, please may I have the first step of the puzzle?

[daj95376]

Aside Note:

Code: Select all

6....2.5952..4..1...35..2..3..1945...1.658.3...5273..1..4..51...3..2..4575.4....8


With respect to Templates, this is the most evenly distributed puzzle I recall encountering. The candidates for 1,2,3,4 appear to be rotationally symmetric to each other. Similarly for the candidates of 6,7,8,9. Maybe that's why it's so difficult to find an elimination.

Code: Select all

~~~~~ Template Entries Remaining

<1> =     4
<2> =     4
<3> =     4
<4> =     4
<5> =     1
<6> =    32
<7> =    32
<8> =    32
<9> =    32


Sometimes, Brute Force has its place. There are four entries for each of the first four candidates. For each candidate, knowing the correct entry leads to a solution with my modest solver.

Code: Select all

6...12.5952..4..1.1.35..2..3..1945...1.658.3...5273..1..4..51...3..21.457514....8
6...12.5952..4..1.1.35..2..3..1945...1.658.3...5273..1..4..51...31.2..4575.4.1..8 *
6.1..2.5952..4..1...35.12..3..1945...1.658.3...5273..1..4..51..13..2..4575.41...8
6.1..2.5952..4..1...351.2..3..1945...1.658.3...5273..1..4..51..13..2..4575.4.1..8


Code: Select all

6....2.5952..4..1...35..2..3..1945.2.12658.3...5273..12.4..51...3..2..4575.4...28
6....2.5952..4..1...35..2..3..1945.221.658.3...5273..1..4..512..3..2..457524....8 *
6....2.5952..4..1...35..2..3..19452.21.658.3...5273..1..4..51.2.3..2..457524....8
6....2.5952..4..1...35..2..3.21945...1.658.32..5273..12.4..51...3..2..4575.4...28


Code: Select all

6....235952.34..1...35..2..3..1945...1.658.3...5273..1..4..51.3.3..2..4575.43...8
6...32.5952..4..13..35..2..3..1945...1.658.3...5273..1..43.51...3..2..4575.4..3.8
6..3.2.5952..4..13..35..2..3..1945...1.658.3...5273..1..4.351...3..2..4575.4..3.8
6..3.2.5952..4.31...35..2..3..1945...1.658.3...5273..1..4..51.3.3..2..4575.43...8 *


Code: Select all

6....245952..4..1..435..2..3..1945...1.658.344.5273..1..4..51...3..2..4575.4....8 *
6....245952..4..1.4.35..2..3..1945...1.658.34.45273..1..4..51...3..2..4575.4....8
64...2.5952..4..1...35..2.43..1945...1.65843.4.5273..1..4..51...3..2..4575.4....8
64...2.5952..4..1...35..2.43..1945..41.658.3...52734.1..4..51...3..2..4575.4....8


[Pat]

in version 1.2.1
it is rated 11.2
= "Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains)"

here's the first step --

Dynamic Contradiction Forcing Chains

With this solving technique, we will prove the two following assertions:

• If R1C7 contains the value 3, then R7C2 must contain the value 6
• If R1C7 contains the value 3, then R7C2 cannot contain the value 6

Because the same assumption yields to contradictory results, we can conclude that the assumption is false, that is, R1C7 cannot contain the value 3.

Each assertion is proved by a different chain of simple rules. The chains can be dynamic, which means that the conclusions of multiple sub-chains must be combined in some cases.

The details of each chain are given below. Use the view selector below the grid to switch between the graphical illustrations of the two different chains.


Chain 1: If R1C7 contains the value 3, then R7C2 cannot contain the value 6 (View 1):
(1) If R1C7 contains the value 3, then R1C7 cannot contain the value 4 (the cell can contain only one value)
(2) If R1C7 does not contain the value 4, then R1C2 must contain the value 4 (only remaining possible position in the row)
(3) If R1C2 contains the value 4, then R1C2 cannot contain the value 7 (the cell can contain only one value)
(4) If R1C7 does not contain the value 4 (1), then R3C9 must contain the value 4 (only remaining possible position in the block)
(5) If R3C9 contains the value 4, then R3C9 cannot contain the value 7 (the cell can contain only one value)
(6) If R1C7 contains the value 3 (initial assumption), then R1C7 cannot contain the value 7 (the cell can contain only one value)
(7) If R1C7 contains the value 3 (initial assumption), then R2C9 cannot contain the value 3 (the value can occur only once in the block)
(8) If R2C9 does not contain the value 3, then R7C9 must contain the value 3 (only remaining possible position in the column)
(9) If R7C9 contains the value 3, then R7C9 cannot contain the value 7 (the cell can contain only one value)
(10) If R7C9 does not contain the value 7, R1C7 does not contain the value 7 (6), R1C2 does not contain the value 7 (3) and R3C9 does not contain the value 7 (5), then R2C3 cannot contain the value 7 (Region Forcing Chains: 7 in row ==> R2C3.7 off)
(11) If R3C9 contains the value 4 (4), then R3C9 cannot contain the value 6 (the cell can contain only one value)
(12) If R7C9 contains the value 3 (8), then R7C9 cannot contain the value 6 (the cell can contain only one value)
(13) If R7C9 contains the value 3 (8), then R7C9 cannot contain the value 2 (the cell can contain only one value)
(14) If R7C9 does not contain the value 2, then R4C8 cannot contain the value 2 (Pointing: Cells R7C8,R9C8: 2 in block and column)
(15) If R1C7 does not contain the value 7 (6), R1C2 does not contain the value 7 (3) and R7C9 does not contain the value 7 (9), then R4C8 cannot contain the value 7 (Forcing X-Chain: R4C8.7 off)
(16) If R4C8 does not contain the value 7, R4C8 does not contain the value 2 (14), R1C2 does not contain the value 7 (3), R7C9 does not contain the value 6 (12) and R3C9 does not contain the value 6 (11), then R4C2 cannot contain the value 6 (Cell Forcing Chains: R3C8 ==> R4C2.6 off)
(17) If R4C8 does not contain the value 7 (15), R4C8 does not contain the value 2 (14), R4C2 does not contain the value 6, R2C3 does not contain the value 7 (10) and R1C2 does not contain the value 7 (3), then R3C1 cannot contain the value 8 (Cell Forcing Chains: R3C8 ==> R3C1.8 off)
(18) If R3C9 contains the value 4 (4), then R3C1 cannot contain the value 4 (the value can occur only once in the row)
(19) If R1C7 contains the value 3 (initial assumption), then R2C7 cannot contain the value 3 (the value can occur only once in the block)
(20) If R2C7 does not contain the value 3 and R2C9 does not contain the value 3 (7), then R2C4 must contain the value 3 (only remaining possible position in the row)
(21) If R2C4 contains the value 3, then R2C4 cannot contain the value 9 (the cell can contain only one value)
(22) If R2C4 does not contain the value 9, then R8C6 cannot contain the value 9 (Pointing: Cells R2C6,R3C6: 9 in block and column)
(23) If R2C4 does not contain the value 9 (21), then R9C6 cannot contain the value 9 (Pointing: Cells R2C6,R3C6: 9 in block and column)
(24) If R9C6 does not contain the value 9, R8C6 does not contain the value 9 (22), R7C9 does not contain the value 7 (9) and R7C9 does not contain the value 2 (13), then R7C1 cannot contain the value 8 (Region Forcing Chains: 8 in block ==> R7C1.8 off)
(25) If R1C7 contains the value 3 (initial assumption), then R1C7 cannot contain the value 8 (the cell can contain only one value)
(26) If R1C7 does not contain the value 8, R2C3 does not contain the value 7 (10) and R7C1 does not contain the value 8 (24), then R3C1 cannot contain the value 9 (Region Forcing Chains: 8 in column ==> R3C1.9 off)
(27) If R3C1 does not contain the value 9, R3C1 does not contain the value 4 (18) and R3C1 does not contain the value 8 (17), then R3C1 must contain the value 1 (only remaining possible value in the cell)
(28) If R3C1 contains the value 1, then R3C5 cannot contain the value 1 (the value can occur only once in the row)
(29) If R1C7 contains the value 3 (initial assumption), then R1C4 cannot contain the value 3 (the value can occur only once in the row)
(30) If R4C8 does not contain the value 7 (15), R4C8 does not contain the value 2 (14), R4C2 does not contain the value 6 (16), R2C3 does not contain the value 7 (10), R1C2 does not contain the value 7 (3) and R1C4 does not contain the value 3, then R3C5 cannot contain the value 8 (Cell Forcing Chains: R3C8 ==> R3C5.8 off)
(31) If R3C5 does not contain the value 8 and R3C5 does not contain the value 1 (28), then R3C5 must contain the value 6 (only remaining possible value in the cell)
(32) If R3C5 contains the value 6, then R7C5 cannot contain the value 6 (the value can occur only once in the column)
(33) If R1C7 does not contain the value 8 (25), R4C8 does not contain the value 7 (15), R4C8 does not contain the value 2 (14), R7C9 does not contain the value 6 (12) and R3C9 does not contain the value 6 (11), then R2C7 cannot contain the value 6 (Forcing Chain: R2C7.6 off)
(34) If R2C7 does not contain the value 6 and R4C2 does not contain the value 6 (16), then R7C8 cannot contain the value 6 (Region Forcing Chains: 6 in column ==> R7C8.6 off)
(35) If R7C9 does not contain the value 6 (12), R7C8 does not contain the value 6 and R7C5 does not contain the value 6 (32), then R7C2 must contain the value 6 (only remaining possible position in the row)


Chain 2: If R7C2 must contain the value 6, then R7C2 cannot contain the value 6 (View 2):
(1) If R1C7 contains the value 3, then R2C9 cannot contain the value 3 (the value can occur only once in the block)
(2) If R2C9 does not contain the value 3, then R7C9 must contain the value 3 (only remaining possible position in the column)
(3) If R7C9 contains the value 3, then R7C9 cannot contain the value 2 (the cell can contain only one value)
(4) If R7C9 contains the value 3 (2), then R7C9 cannot contain the value 7 (the cell can contain only one value)
(5) If R1C7 contains the value 3 (initial assumption), then R1C7 cannot contain the value 4 (the cell can contain only one value)
(6) If R1C7 does not contain the value 4, then R1C2 must contain the value 4 (only remaining possible position in the row)
(7) If R1C2 contains the value 4, then R1C2 cannot contain the value 7 (the cell can contain only one value)
(8) If R1C7 contains the value 3 (initial assumption), then R1C7 cannot contain the value 7 (the cell can contain only one value)
(9) If R1C7 does not contain the value 7, R1C2 does not contain the value 7 (7) and R7C9 does not contain the value 7 (4), then R4C8 cannot contain the value 7 (Forcing X-Chain: R4C8.7 off)
(10) If R1C7 contains the value 3 (initial assumption), then R2C7 cannot contain the value 3 (the value can occur only once in the block)
(11) If R1C7 does not contain the value 4 (5), then R3C9 must contain the value 4 (only remaining possible position in the block)
(12) If R3C9 contains the value 4, then R3C9 cannot contain the value 6 (the cell can contain only one value)
(13) If R7C9 contains the value 3 (2), then R7C9 cannot contain the value 6 (the cell can contain only one value)
(14) If R7C9 does not contain the value 2 (3), then R4C8 cannot contain the value 2 (Pointing: Cells R7C8,R9C8: 2 in block and column)
(15) If R1C7 contains the value 3 (initial assumption), then R1C7 cannot contain the value 8 (the cell can contain only one value)
(16) If R1C7 does not contain the value 8, R4C8 does not contain the value 7 (9), R4C8 does not contain the value 2 (14), R7C9 does not contain the value 6 (13) and R3C9 does not contain the value 6 (12), then R2C7 cannot contain the value 6 (Forcing Chain: R2C7.6 off)
(17) If R3C9 contains the value 4 (11), then R3C9 cannot contain the value 7 (the cell can contain only one value)
(18) If R7C9 does not contain the value 7 (4), R1C7 does not contain the value 7 (8), R1C2 does not contain the value 7 (7) and R3C9 does not contain the value 7, then R2C3 cannot contain the value 7 (Region Forcing Chains: 7 in row ==> R2C3.7 off)
(19) If R2C3 does not contain the value 7, R2C7 does not contain the value 6 (16), R2C7 does not contain the value 3 (10), R1C2 does not contain the value 7 (7), R4C8 does not contain the value 7 (9) and R7C9 does not contain the value 2 (3), then R9C8 cannot contain the value 9 (Cell Forcing Chains: R5C3 ==> R9C8.2 on)
(20) If R2C7 does not contain the value 3 (10) and R2C9 does not contain the value 3 (1), then R2C4 must contain the value 3 (only remaining possible position in the row)
(21) If R2C4 contains the value 3, then R2C4 cannot contain the value 9 (the cell can contain only one value)
(22) If R2C4 does not contain the value 9, then R9C6 cannot contain the value 9 (Pointing: Cells R2C6,R3C6: 9 in block and column)
(23) If R9C6 does not contain the value 9, R2C3 does not contain the value 7 (18), R2C7 does not contain the value 6 (16), R2C7 does not contain the value 3 (10) and R4C8 does not contain the value 7 (9), then R7C8 cannot contain the value 9 (Region Forcing Chains: 9 in row ==> R7C8.9 off)
(24) If R7C8 does not contain the value 9 and R9C8 does not contain the value 9 (19), then R6C8 must contain the value 9 (only remaining possible position in the column)
(25) If R6C8 contains the value 9, then R6C2 cannot contain the value 9 (the value can occur only once in the row)
(26) If R1C2 contains the value 4 (6), then R6C2 cannot contain the value 4 (the value can occur only once in the column)
(27) If R4C8 does not contain the value 7 (9), R4C8 does not contain the value 2 (14), R1C2 does not contain the value 7 (7), R7C9 does not contain the value 6 (13) and R3C9 does not contain the value 6 (12), then R4C2 cannot contain the value 6 (Cell Forcing Chains: R3C8 ==> R4C2.6 off)
(28) If R3C9 contains the value 4 (11), then R3C2 cannot contain the value 4 (the value can occur only once in the row)
(29) If R2C3 does not contain the value 7 (18), R1C7 does not contain the value 8 (15), R3C2 does not contain the value 4 and R4C2 does not contain the value 6 (27), then R6C2 cannot contain the value 8 (Cell Forcing Chains: R3C2 ==> R6C2.8 off)
(30) If R6C2 does not contain the value 8, R6C2 does not contain the value 4 (26) and R6C2 does not contain the value 9 (25), then R6C2 must contain the value 6 (only remaining possible value in the cell)
(31) If R6C2 contains the value 6, then R7C2 cannot contain the value 6 (the value can occur only once in the column)




Nested Forcing Chains details (Note that each Nested Forcing Chain relies on the fact that some candidates have been excluded by the main Forcing Chain):


Nested Region Forcing Chains: 6 in column ==> R7C8.6 off

Chain 3: If R6C7 contains the value 6, then R7C8 cannot contain the value 6 (View 3):
(1) If R6C7 contains the value 6, then R6C2 cannot contain the value 6 (the value can occur only once in the row)
(2) If R6C2 does not contain the value 6, then R7C2 must contain the value 6 (only remaining possible position in the column)
(3) If R7C2 contains the value 6, then R7C8 cannot contain the value 6 (the value can occur only once in the row)

Chain 4: If R8C7 contains the value 6, then R7C8 cannot contain the value 6 (View 4):
(1) If R8C7 contains the value 6, then R7C8 cannot contain the value 6 (the value can occur only once in the block)

Chain 5: If R9C7 contains the value 6, then R7C8 cannot contain the value 6 (View 5):
(1) If R9C7 contains the value 6, then R7C8 cannot contain the value 6 (the value can occur only once in the block)


Nested Cell Forcing Chains: R3C8 ==> R4C2.6 off

Chain 6: If R3C8 contains the value 6, then R4C2 cannot contain the value 6 (View 6):
(1) If R3C8 contains the value 6, then R2C9 cannot contain the value 6 (the value can occur only once in the block)
(2) If R2C9 does not contain the value 6, then R4C9 must contain the value 6 (only remaining possible position in the column)
(3) If R4C9 contains the value 6, then R4C2 cannot contain the value 6 (the value can occur only once in the row)

Chain 7: If R3C8 contains the value 7, then R4C2 cannot contain the value 6 (View 7):
(1) If R3C8 contains the value 7, then R3C2 cannot contain the value 7 (the value can occur only once in the row)
(2) If R3C2 does not contain the value 7, then R4C2 must contain the value 7 (only remaining possible position in the column)
(3) If R4C2 contains the value 7, then R4C2 cannot contain the value 6 (the cell can contain only one value)

Chain 8: If R3C8 contains the value 8, then R4C2 cannot contain the value 6 (View 8):
(1) If R3C8 contains the value 8, then R4C8 cannot contain the value 8 (the value can occur only once in the column)
(2) If R4C8 does not contain the value 8, then R4C8 must contain the value 6 (only remaining possible value in the cell)
(3) If R4C8 contains the value 6, then R4C2 cannot contain the value 6 (the value can occur only once in the row)


Nested Forcing Chain: R2C7.6 off

Chain 9: If R2C7 contains the value 6, then R2C7 cannot contain the value 6 (View 9):
(1) If R2C7 contains the value 6, then R2C7 cannot contain the value 8 (the cell can contain only one value)
(2) If R2C7 does not contain the value 8, then R3C8 must contain the value 8 (only remaining possible position in the block)
(3) If R3C8 contains the value 8, then R4C8 cannot contain the value 8 (the value can occur only once in the column)
(4) If R4C8 does not contain the value 8, then R4C8 must contain the value 6 (only remaining possible value in the cell)
(5) If R4C8 contains the value 6, then R4C9 cannot contain the value 6 (the value can occur only once in the block)
(6) If R4C9 does not contain the value 6, then R2C9 must contain the value 6 (only remaining possible position in the column)
(7) If R2C9 contains the value 6, then R2C7 cannot contain the value 6 (the value can occur only once in the block)


Nested Cell Forcing Chains: R3C8 ==> R3C5.8 off

Chain 10: If R3C8 contains the value 6, then R3C5 cannot contain the value 8 (View 10):
(1) If R3C8 contains the value 6, then R4C8 cannot contain the value 6 (the value can occur only once in the column)
(2) If R4C8 does not contain the value 6, then R4C8 must contain the value 8 (only remaining possible value in the cell)
(3) If R4C8 contains the value 8, then R4C2 cannot contain the value 8 (the value can occur only once in the row)
(4) If R4C2 does not contain the value 8, then R4C2 must contain the value 7 (only remaining possible value in the cell)
(5) If R4C2 contains the value 7, then R3C2 cannot contain the value 7 (the value can occur only once in the column)
(6) If R3C2 does not contain the value 7, then R1C3 must contain the value 7 (only remaining possible position in the block)
(7) If R1C3 contains the value 7, then R1C4 cannot contain the value 7 (the value can occur only once in the row)
(8) If R1C4 does not contain the value 7, then R1C4 must contain the value 8 (only remaining possible value in the cell)
(9) If R1C4 contains the value 8, then R3C5 cannot contain the value 8 (the value can occur only once in the block)

Chain 11: If R3C8 contains the value 7, then R3C5 cannot contain the value 8 (View 11):
(1) If R3C8 contains the value 7, then R3C2 cannot contain the value 7 (the value can occur only once in the row)
(2) If R3C2 does not contain the value 7, then R1C3 must contain the value 7 (only remaining possible position in the block)
(3) If R1C3 contains the value 7, then R1C4 cannot contain the value 7 (the value can occur only once in the row)
(4) If R1C4 does not contain the value 7, then R1C4 must contain the value 8 (only remaining possible value in the cell)
(5) If R1C4 contains the value 8, then R3C5 cannot contain the value 8 (the value can occur only once in the block)

Chain 12: If R3C8 contains the value 8, then R3C5 cannot contain the value 8 (View 12):
(1) If R3C8 contains the value 8, then R3C5 cannot contain the value 8 (the value can occur only once in the row)


Nested Forcing X-Chain: R4C8.7 off

Chain 13: If R4C8 contains the value 7, then R4C8 cannot contain the value 7 (View 13):
(1) If R4C8 contains the value 7, then R4C2 cannot contain the value 7 (the value can occur only once in the row)
(2) If R4C2 does not contain the value 7, then R3C2 must contain the value 7 (only remaining possible position in the column)
(3) If R3C2 contains the value 7, then R1C3 cannot contain the value 7 (the value can occur only once in the block)
(4) If R1C3 does not contain the value 7, then R1C4 must contain the value 7 (only remaining possible position in the row)
(5) If R1C4 contains the value 7, then R7C4 cannot contain the value 7 (the value can occur only once in the column)
(6) If R7C4 does not contain the value 7, then R7C8 must contain the value 7 (only remaining possible position in the row)
(7) If R7C8 contains the value 7, then R4C8 cannot contain the value 7 (the value can occur only once in the column)


Nested Region Forcing Chains: 7 in row ==> R2C3.7 off

Chain 14: If R3C2 contains the value 7, then R2C3 cannot contain the value 7 (View 14):
(1) If R3C2 contains the value 7, then R2C3 cannot contain the value 7 (the value can occur only once in the block)

Chain 15: If R3C6 contains the value 7, then R2C3 cannot contain the value 7 (View 15):
(1) If R3C6 contains the value 7, then R1C4 cannot contain the value 7 (the value can occur only once in the block)
(2) If R1C4 does not contain the value 7, then R1C3 must contain the value 7 (only remaining possible position in the row)
(3) If R1C3 contains the value 7, then R2C3 cannot contain the value 7 (the value can occur only once in the block)

Chain 16: If R3C8 contains the value 7, then R2C3 cannot contain the value 7 (View 16):
(1) If R3C8 contains the value 7, then R7C8 cannot contain the value 7 (the value can occur only once in the column)
(2) If R7C8 does not contain the value 7, then R7C4 must contain the value 7 (only remaining possible position in the row)
(3) If R7C4 contains the value 7, then R1C4 cannot contain the value 7 (the value can occur only once in the column)
(4) If R1C4 does not contain the value 7, then R1C3 must contain the value 7 (only remaining possible position in the row)
(5) If R1C3 contains the value 7, then R2C3 cannot contain the value 7 (the value can occur only once in the block)


Nested Cell Forcing Chains: R3C8 ==> R3C1.8 off

Chain 17: If R3C8 contains the value 6, then R3C1 cannot contain the value 8 (View 17):
(1) If R3C8 contains the value 6, then R4C8 cannot contain the value 6 (the value can occur only once in the column)
(2) If R4C8 does not contain the value 6, then R4C8 must contain the value 8 (only remaining possible value in the cell)
(3) If R4C8 contains the value 8, then R4C2 cannot contain the value 8 (the value can occur only once in the row)
(4) If R4C2 does not contain the value 8, then R4C2 must contain the value 7 (only remaining possible value in the cell)
(5) If R4C2 contains the value 7, then R3C2 cannot contain the value 7 (the value can occur only once in the column)
(6) If R3C2 does not contain the value 7, then R1C3 must contain the value 7 (only remaining possible position in the block)
(7) If R1C3 contains the value 7, then R1C3 cannot contain the value 1 (the cell can contain only one value)
(8) If R1C3 does not contain the value 1, then R3C1 must contain the value 1 (only remaining possible position in the block)
(9) If R3C1 contains the value 1, then R3C1 cannot contain the value 8 (the cell can contain only one value)

Chain 18: If R3C8 contains the value 7, then R3C1 cannot contain the value 8 (View 18):
(1) If R3C8 contains the value 7, then R3C2 cannot contain the value 7 (the value can occur only once in the row)
(2) If R3C2 does not contain the value 7, then R1C3 must contain the value 7 (only remaining possible position in the block)
(3) If R1C3 contains the value 7, then R1C3 cannot contain the value 1 (the cell can contain only one value)
(4) If R1C3 does not contain the value 1, then R3C1 must contain the value 1 (only remaining possible position in the block)
(5) If R3C1 contains the value 1, then R3C1 cannot contain the value 8 (the cell can contain only one value)

Chain 19: If R3C8 contains the value 8, then R3C1 cannot contain the value 8 (View 19):
(1) If R3C8 contains the value 8, then R3C1 cannot contain the value 8 (the value can occur only once in the row)


Nested Region Forcing Chains: 8 in column ==> R3C1.9 off

Chain 20: If R3C1 contains the value 8, then R3C1 cannot contain the value 9 (View 20):
(1) If R3C1 contains the value 8, then R3C1 cannot contain the value 9 (the cell can contain only one value)

Chain 21: If R6C1 contains the value 8, then R3C1 cannot contain the value 9 (View 21):
(1) If R6C1 contains the value 8, then R6C7 cannot contain the value 8 (the value can occur only once in the row)
(2) If R6C7 does not contain the value 8, then R2C7 must contain the value 8 (only remaining possible position in the column)
(3) If R2C7 contains the value 8, then R2C3 cannot contain the value 8 (the value can occur only once in the row)
(4) If R2C3 does not contain the value 8, then R2C3 must contain the value 9 (only remaining possible value in the cell)
(5) If R2C3 contains the value 9, then R3C1 cannot contain the value 9 (the value can occur only once in the block)

Chain 22: If R8C1 contains the value 8, then R3C1 cannot contain the value 9 (View 22):
(1) If R8C1 contains the value 8, then R8C1 cannot contain the value 1 (the cell can contain only one value)
(2) If R8C1 does not contain the value 1, then R3C1 must contain the value 1 (only remaining possible position in the column)
(3) If R3C1 contains the value 1, then R3C1 cannot contain the value 9 (the cell can contain only one value)


Nested Region Forcing Chains: 8 in block ==> R7C1.8 off

Chain 23: If R7C4 contains the value 8, then R7C1 cannot contain the value 8 (View 23):
(1) If R7C4 contains the value 8, then R7C1 cannot contain the value 8 (the value can occur only once in the row)

Chain 24: If R7C5 contains the value 8, then R7C1 cannot contain the value 8 (View 24):
(1) If R7C5 contains the value 8, then R7C1 cannot contain the value 8 (the value can occur only once in the row)

Chain 25: If R8C4 contains the value 8, then R7C1 cannot contain the value 8 (View 25):
(1) If R8C4 contains the value 8, then R8C4 cannot contain the value 9 (the cell can contain only one value)
(2) If R8C4 does not contain the value 9, then R7C4 must contain the value 9 (only remaining possible position in the block)
(3) If R7C4 contains the value 9, then R7C4 cannot contain the value 7 (the cell can contain only one value)
(4) If R7C4 does not contain the value 7, then R7C8 must contain the value 7 (only remaining possible position in the row)
(5) If R7C8 contains the value 7, then R7C8 cannot contain the value 2 (the cell can contain only one value)
(6) If R7C8 does not contain the value 2, then R7C1 must contain the value 2 (only remaining possible position in the row)
(7) If R7C1 contains the value 2, then R7C1 cannot contain the value 8 (the cell can contain only one value)


Nested Cell Forcing Chains: R3C2 ==> R6C2.8 off

Chain 26: If R3C2 contains the value 7, then R6C2 cannot contain the value 8 (View 26):
(1) If R3C2 contains the value 7, then R4C2 cannot contain the value 7 (the value can occur only once in the column)
(2) If R4C2 does not contain the value 7, then R4C2 must contain the value 8 (only remaining possible value in the cell)
(3) If R4C2 contains the value 8, then R6C2 cannot contain the value 8 (the value can occur only once in the block)

Chain 27: If R3C2 contains the value 8, then R6C2 cannot contain the value 8 (View 27):
(1) If R3C2 contains the value 8, then R6C2 cannot contain the value 8 (the value can occur only once in the column)

Chain 28: If R3C2 contains the value 9, then R6C2 cannot contain the value 8 (View 28):
(1) If R3C2 contains the value 9, then R2C3 cannot contain the value 9 (the value can occur only once in the block)
(2) If R2C3 does not contain the value 9, then R2C3 must contain the value 8 (only remaining possible value in the cell)
(3) If R2C3 contains the value 8, then R2C7 cannot contain the value 8 (the value can occur only once in the row)
(4) If R2C7 does not contain the value 8, then R6C7 must contain the value 8 (only remaining possible position in the column)
(5) If R6C7 contains the value 8, then R6C2 cannot contain the value 8 (the value can occur only once in the row)


Nested Cell Forcing Chains: R5C3 ==> R9C8.2 on

Chain 29: If R5C3 contains the value 2, then R9C8 must contain the value 2 (View 29):
(1) If R5C3 contains the value 2, then R9C3 cannot contain the value 2 (the value can occur only once in the column)
(2) If R9C3 does not contain the value 2, then R9C8 must contain the value 2 (only remaining possible position in the row)

Chain 30: If R5C3 contains the value 7, then R9C8 must contain the value 2 (View 30):
(1) If R5C3 contains the value 7, then R4C2 cannot contain the value 7 (the value can occur only once in the block)
(2) If R4C2 does not contain the value 7, then R3C2 must contain the value 7 (only remaining possible position in the column)
(3) If R3C2 contains the value 7, then R3C8 cannot contain the value 7 (the value can occur only once in the row)
(4) If R3C8 does not contain the value 7, then R7C8 must contain the value 7 (only remaining possible position in the column)
(5) If R7C8 contains the value 7, then R7C8 cannot contain the value 2 (the cell can contain only one value)
(6) If R7C8 does not contain the value 2, then R9C8 must contain the value 2 (only remaining possible position in the block)

Chain 31: If R5C3 contains the value 9, then R9C8 must contain the value 2 (View 31):
(1) If R5C3 contains the value 9, then R2C3 cannot contain the value 9 (the value can occur only once in the column)
(2) If R2C3 does not contain the value 9, then R2C3 must contain the value 8 (only remaining possible value in the cell)
(3) If R2C3 contains the value 8, then R2C7 cannot contain the value 8 (the value can occur only once in the row)
(4) If R2C7 does not contain the value 8, then R2C7 must contain the value 7 (only remaining possible value in the cell)
(5) If R2C7 contains the value 7, then R3C8 cannot contain the value 7 (the value can occur only once in the block)
(6) If R3C8 does not contain the value 7, then R7C8 must contain the value 7 (only remaining possible position in the column)
(7) If R7C8 contains the value 7, then R7C8 cannot contain the value 2 (the cell can contain only one value)
(8) If R7C8 does not contain the value 2, then R9C8 must contain the value 2 (only remaining possible position in the block)


Nested Region Forcing Chains: 9 in row ==> R7C8.9 off

Chain 32: If R9C3 contains the value 9, then R7C8 cannot contain the value 9 (View 32):
(1) If R9C3 contains the value 9, then R2C3 cannot contain the value 9 (the value can occur only once in the column)
(2) If R2C3 does not contain the value 9, then R2C3 must contain the value 8 (only remaining possible value in the cell)
(3) If R2C3 contains the value 8, then R2C7 cannot contain the value 8 (the value can occur only once in the row)
(4) If R2C7 does not contain the value 8, then R2C7 must contain the value 7 (only remaining possible value in the cell)
(5) If R2C7 contains the value 7, then R3C8 cannot contain the value 7 (the value can occur only once in the block)
(6) If R3C8 does not contain the value 7, then R7C8 must contain the value 7 (only remaining possible position in the column)
(7) If R7C8 contains the value 7, then R7C8 cannot contain the value 9 (the cell can contain only one value)

Chain 33: If R9C7 contains the value 9, then R7C8 cannot contain the value 9 (View 33):
(1) If R9C7 contains the value 9, then R7C8 cannot contain the value 9 (the value can occur only once in the block)

Chain 34: If R9C8 contains the value 9, then R7C8 cannot contain the value 9 (View 34):
(1) If R9C8 contains the value 9, then R7C8 cannot contain the value 9 (the value can occur only once in the block)