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. . . | 9 . 8 | . 6 .
3 . . | . . . | 2 . .
. . 1 | . 2 . | . . .
-------+-------+------
. 8 . | 1 . . | . 9 .
. . 2 | . . . | 3 . .
. 7 . | . . 5 | . 4 .
-------+-------+------
. . . | . 4 . | 6 . .
. . 5 | . . . | . . 1
. 9 . | 6 . 7 | . . .
Standard techniques take us so far:
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2|4|5|7 2|4|5 4|7 | 9 1|3|5 8 | 1|5 6 3|4|5
3 4|5|6 9 | 4|5|7 5|6|7 1|4|6 | 2 1|5|7|8 4|5|7|8
8 4|5|6 1 | 3|4|5|7 2 3|4|6 | 4|5|7|9 3|5|7 3|4|5|7|9
----------------------+----------------------------+-----------------------------
4|5 8 3|6 | 1 3|6|7 2|3|4|6 | 5|7 9 2|5|6|7
4|5|9 1|4|5 2 | 4|7|8 6|7|8|9 4|6|9 | 3 1|5|7|8 5|6|7|8
1|9 7 3|6 | 2|3|8 3|6|8|9 5 | 1|8 4 2|6|8
----------------------+----------------------------+-----------------------------
2|7 1|2|3 7|8 | 2|3|5|8 4 1|2|3|9 | 6 2|3|5|8 3|5|8|9
6 2|3|4 5 | 2|3|8 3|8|9 2|3|9 | 4|7|8|9 2|3|7|8 1
1|2|4 9 4|8 | 6 1|3|5|8 7 | 4|5|8 2|3|5|8 3|4|5|8
whereupon we note:
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Consider the chain r7c6-1-r7c2-3-r8c2=<4|9>=r7c6.
When the cell r7c6 contains the value 9, it likewise contains the value 1 - a contradiction.
Therefore, the cell r7c6 cannot contain the value 9.
- The move r7c6:=9 has been eliminated.
The cell r7c9 is the only candidate for the value 9 in Row 7.
The point is that the cells {r8c2,r8c4,r8c5,r8c6} form an Almost Locked Set, so when r7c6 takes the value 9, the set is locked, which forces r8c2 to take the value 4. Of course, we could have solved the puzzle without the link but the resulting chain is very short and the logic necessary to follow the link falls well within 'human' capabilities.
After a couple more observations:
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2. The cell r3c7 is the only candidate for the value 9 in Row 3.
3. The value 4 in Box 9 must lie in Column 7.
- The move r9c9:=4 has been eliminated.
2|4|5|7 2|4|5 4|7 | 9 1|3|5 8 | 1|5 6 3|4|5
3 4|5|6 9 | 4|5|7 5|6|7 1|4|6 | 2 1|5|7|8 4|5|7|8
8 4|5|6 1 | 3|4|5|7 2 3|4|6 | 9 3|5|7 3|4|5|7
----------------------+----------------------------+-------------------------
4|5 8 3|6 | 1 3|6|7 2|3|4|6 | 5|7 9 2|5|6|7
4|5|9 1|4|5 2 | 4|7|8 6|7|8|9 4|6|9 | 3 1|5|7|8 5|6|7|8
1|9 7 3|6 | 2|3|8 3|6|8|9 5 | 1|8 4 2|6|8
----------------------+----------------------------+-------------------------
2|7 1|2|3 7|8 | 2|3|5|8 4 1|2|3 | 6 2|3|5|8 9
6 2|3|4 5 | 2|3|8 3|8|9 2|3|9 | 4|7|8 2|3|7|8 1
1|2|4 9 4|8 | 6 1|3|5|8 7 | 4|5|8 2|3|5|8 3|5|8
we find another link:
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Consider the chain r7c8=<5|3>=r9c9~3~r1c9-3-r1c5-1-r9c5=5=r7c8.
When the cell r7c8 contains the value 5, the chain is self-contradicting.
Therefore, the cell r7c8 cannot contain the value 5.
- The move r7c8:=5 has been eliminated.
The cell r7c4 is the only candidate for the value 5 in Row 7.
Here an Almost Locked Set is formed by the cells {r9c3,r9c7,r9c9}, so when r7c8 takes a 5, r9c9 must take a 3. The final extended link works because when r9c5 doesn't take a 5, one of {r9c7,r9c8,r9c9} must, which means that r7c8 doesn't.
Yet another extended disjoint subset link has to be found in order to progress beyond the last elimination. (It's seldom necessary to use these links but this puzzle provides an unusual number of examples).
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2|4|5|7 2|4|5 4|7 | 9 1|3|5 8 | 1|5 6 3|4|5
3 4|5|6 9 | 4|7 5|6|7 1|4|6 | 2 1|5|7|8 4|5|7|8
8 4|5|6 1 | 3|4|7 2 3|4|6 | 9 3|5|7 3|4|5|7
----------------------+--------------------------+-------------------------
4|5 8 3|6 | 1 3|6|7 2|3|4|6 | 5|7 9 2|5|6|7
4|5|9 1|4|5 2 | 4|7|8 6|7|8|9 4|6|9 | 3 1|5|7|8 5|6|7|8
1|9 7 3|6 | 2|3|8 3|6|8|9 5 | 1|8 4 2|6|8
----------------------+--------------------------+-------------------------
2|7 1|2|3 7|8 | 5 4 1|2|3 | 6 2|3|8 9
6 2|3|4 5 | 2|3|8 3|8|9 2|3|9 | 4|7|8 2|3|7|8 1
1|2|4 9 4|8 | 6 1|3|8 7 | 4|5|8 2|3|5|8 3|5|8
Consider the chain r9c5=<8|4>=r8c2~4~r9c3~8~r9c5.
When the cell r9c5 contains the value 8, the chain is self-contradicting.
Therefore, the cell r9c5 cannot contain the value 8.
The following puzzles are left as (fairly difficult) exercises:
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I.
. . 3 | . . 6 | . . .
. 8 . | . . . | . 9 .
. . 7 | . 2 . | . . 4
-------+-------+------
. 9 . | . . 8 | . . 5
. . 4 | . . . | 6 . .
2 . . | 1 . . | . 7 .
-------+-------+------
7 . . | . 4 . | 2 . .
. 5 . | . . . | . 3 .
. . . | 9 . . | 1 . .
II.
. . 3 | . 6 . | . . 2
. . 5 | . . 7 | . . .
6 . . | . . . | . 1 .
-------+-------+------
. . 1 | . . 5 | . 4 .
. 6 . | . . . | . 3 .
. 8 . | 1 . . | 9 . .
-------+-------+------
. 9 . | . . . | . . 8
. . . | 4 . . | 3 . .
7 . . | . 2 . | 6 . .
III.
. 3 . | 9 . . | . . .
2 . . | 7 . . | . . 6
. . 1 | . . 6 | . 5 .
-------+-------+------
. . 9 | . . . | 2 . .
8 . . | . . . | . . 4
. . 5 | . . . | 3 . .
-------+-------+------
. 4 . | 2 . . | 9 . .
6 . . | . . 8 | . . 7
. . . | . . 5 | . 1 .
IV.
. . . | 4 . . | . . 8
. 7 5 | . . . | 9 . .
. . 2 | . . 7 | . 1 .
-------+-------+------
. . 8 | . . 2 | . . .
6 . . | . . . | . . 3
. . . | 9 . . | 5 . .
-------+-------+------
. 9 . | 7 . . | 1 . .
. . 1 | . . . | 8 2 .
3 . . | . . 6 | . . .
