- Code: Select all
` 2 9 5 | 7 . . | 8 6 1 `

. 3 1 | 8 6 5 | 9 2 .

8 . 6 | . . . | . . .

-------+-------+------

. . 7 | . 5 . | . . 6

. . . | 3 8 7 | . . .

5 . . | . 1 6 | 7 . .

-------+-------+------

. . . | 5 . . | 1 . 9

. 2 . | 6 . . | 3 5 .

. 5 4 | . . 8 | 6 7 2

- Code: Select all
` `

2 9 5 | 7 34 34 | 8 6 1

47 3 1 | 8 6 5 | 9 2 47

8 47 6 | 1249 2349 12349 | 45 34 3457

---------------------+----------------------+-------------------

1349 148 7 | 249 5 249 | 24 13489 6

1469 146 29 | 3 8 7 | 245 149 45

5 48 2389 | 249 1 6 | 7 3489 348

---------------------+----------------------+-------------------

367 678 38 | 5 2347 234 | 1 48 9

179 2 89 | 6 479 149 | 3 5 48

139 5 4 | 19 39 8 | 6 7 2

1) r1c56=[34][34]; this is a "naked pair"; it eliminates other 3's and 4's from the rest of box 2.

2) After (1), r46c4 are the only two cells in column 4 that can hold a 4; This is "locked candidates"; eliminate the other 4 from box 5

3) r78c1 are the only two cells that can hold a 1 in box 7. This are "locked candidates"; elimiate other 1's from column 1

4) r4c7+r5c7+r5c9=[24][245][45]; this is a "naked triple"; it eliminates other 2s, 4s and 5s from box 6.

5) r4c467=[249][249][24]; another "naked triple"; it eliminates other 2s, 4s and 9s from from row 4.

After these eliminations, several singles are possible. Updating the grid:

- Code: Select all
` 2 9 5 | 7 4 3 | 8 6 1 `

. 3 1 | 8 6 5 | 9 2 .

8 . 6 | . . . | . . .

-------+-------+------

3 . 7 | . 5 . | . . 6

. . . | 3 8 7 | . . .

5 . . | . 1 6 | 7 . .

-------+-------+------

. . 3 | 5 . . | 1 . 9

. 2 . | 6 . . | 3 5 .

. 5 4 | . 3 8 | 6 7 2

- Code: Select all
`2 9 5 | 7 4 3 | 8 6 1 `

47 3 1 | 8 6 5 | 9 2 47

8 47 6 | 129 29 129 | 45 34 3457

------------------+-------------------+-----------------

3 *18 7 | 249 5 29 | 24 *18 6

469 146 29 | 3 8 7 | 245 19 45

5 x48 289 | 249 1 6 | 7 x389 38

------------------+-------------------+-----------------

67 *678 3 | 5 27 24 | 1 *48 9

179 2 89 | 6 79 149 | 3 5 48

19 5 4 | 19 3 8 | 6 7 2

6) There are only two places for a 9 in row 4, both of which are in box 5. This is locked candidates. The other 4 in box 5 is excluded.

7) There are only two cells in ROWS 4 and 7 that can hold an 8 (marked above with '*') -- all are in COLUMNS 2 and 8. This is an "x-wing". All other 8s in those two COLUMNS are eliminated (marked above with 'x')

After these eliminations, several singles are possible. Updating the grid:

- Code: Select all
` 2 9 5 | 7 4 3 | 8 6 1 `

4 3 1 | 8 6 5 | 9 2 7

8 7 6 | . . . | . . .

-------+-------+------

3 . 7 | 4 5 9 | 2 . 6

. . 2 | 3 8 7 | . . .

5 4 . | 2 1 6 | 7 . .

-------+-------+------

. . 3 | 5 . . | 1 . 9

. 2 . | 6 . . | 3 5 .

. 5 4 | . 3 8 | 6 7 2

- Code: Select all
`2 9 5 | 7 4 3 | 8 6 1 `

4 3 1 | 8 6 5 | 9 2 7

8 7 6 | 19 29 12 | 45 34 345

---------------+----------------+--------------

3 18 7 | 4 5 9 | 2 18 6

69 16 2 | 3 8 7 | 45 19 45

5 4 89 | 2 1 6 | 7 39 38

---------------+----------------+--------------

67 68 3 | 5 27 24 | 1 48 9

179 2 89 | 6 79 14 | 3 5 48

19 5 4 | 19 3 8 | 6 7 2

There are lots of fun ways to finish off the puzzle. Which to use is a matter of taste.

The four cells r35c79 form a UNIQUE RECTANGLE. If r3c9 was [45] the puzzle would have multiple soltions. This is not allowed. Therefore, r3c9=3. After this, the rest of the puzzle is singles.

OR

r8c1+r8c5+r9c1 form an XYZ-wing. No matter WHAT value you place in ANY one of these three cells, the immediate result is that r8c3<>9. Therefore, r8c3=8 and the rest of the puzzle is just singles.

OR

8c) There are dozens of simple xy-type forcing chains of various lengths, all of which will solve the puzzle. For example, here's a five cell chain:

r9c1=1 -> r9c4=9

r9c1=9 -> r8c3=8 -> r8c9=4 -> r8c6=1 -> r9c4=9

Therefore, r9c4=9

This can be expressed as a nice loop:

[r9c1]-9-[r8c3]-8-[r8c9]-4-[r8c5]-1-[r9c4]-9-[r9c1] --> r9c1<>1

OR

There only two unsolved cells with three candidates, the rest have only two. The 'tri-value' cells are r8c1 and r3c9. If r8c1 were [17] instead of [179] and r3c9 were [35] instead of [345], the puzzle would have mutliple solutions (this is BUG -- bivalue universal grave). Therefore, either r8c1=9 OR r3c9=4 OR both.

This allows another 5 cell forcing chain using BUG avoidance as one link:

r8c9=8 -> r8c2=9 -> r8c5=7

r8c9=4 -> r3c9<>4 -> r8c9=9 -> r8c5=7

Therefore, r8c5=7