Hi Yogi. There are many ways to solve this puzzle from the second position, but I've chosen a solution path that only requires one type of move that I assume you are not familiar with.

I'll explain the first move in detail and outline the rest of the solution.

An ALS (Almost Locked Set) is a group of N cells with a total of N+1 different digits. The trick is that if any one of the digits is False in the set, then all other digits must be True in the set.

From your position ....23...1...972...7..84.6...82...93.6.349.7.93.8..62..9.732.8...59....2.2.4..... there are two basic moves which I assume that you are familiar with :

Pointing Pair of 5's r9c56 which removes the 5's from r9c789.

Box column intersection (one of r12c8 must be 5, which removes rest of the 5's from Box 3. This should get you to here :

- Code: Select all
`*--------------------------------------------------------------------------------*`

| 4568 458 469 | 156 2 3 | 14789 145 14789 |

| 1 a458 346 | 6-5 9 7 | 2 a345 a48 |

| 235 7 239 |b15 8 4 |b139 6 b19 |

|--------------------------+--------------------------+--------------------------|

| 457 145 8 | 2 1567 156 | 145 9 3 |

| 25 6 12 | 3 4 9 | 158 7 158 |

| 9 3 147 | 8 157 15 | 6 2 145 |

|--------------------------+--------------------------+--------------------------|

| 46 9 146 | 7 3 2 | 145 8 1456 |

| 34678 148 5 | 9 16 168 | 1347 134 2 |

| 3678 2 1367 | 4 156 1568 | 1379 13 1679 |

*--------------------------------------------------------------------------------*

Now look at the 3 cells I've marked a in the PM diagram. They are an ALS because they contain a total of 4 different digits 3458. Now suppose that both of the 5's in those cells was False in the set. Then those cells must contain 348, in particular r2c8 must be 3. Now look at the 3 cells I've marked as b in the PM diagram. They are an ALS and contain 4 different digits 1359. Now suppose 3 was False in the set (because r2c8 was 3). Then it must contain 159, in particular r3c4 must be 5.

So, summarizing all this, if 5 r2c28 is false => 5 r3c4 must be True. You can reverse this argument and suppose that 5 r3c4 was False and conclude that 5 r3c28 must be True. The net result of all this is that at least one of these 3 cells must be 5.

Now look at r2c4. It can see all of these 3 cells, so it can't be 5 !

The notation of all this is ALS XZ Rule: X = 3, Z = 5: (5=3) r2c289 - (3=5) r3c479 => - 5 r2c4. Z is the pincer digit (elimination digit) and X is the digit that links the 2 ALSs (also called the restricted common digit).

The next move follows on immediately and is similar:

- Code: Select all
`*--------------------------------------------------------------------------------*`

|b4568 b458 469 |b15 2 3 |b14789 b145 b14789 |

| 1 458 34 | 6 9 7 | 2 345 a48 |

| 235 7 239 | 15 8 4 | 139 6 a19 |

|--------------------------+--------------------------+--------------------------|

| 457 145 8 | 2 1567 156 | 145 9 3 |

| 25 6 12 | 3 4 9 | 158 7 a158 |

| 9 3 147 | 8 157 15 | 6 2 a145 |

|--------------------------+--------------------------+--------------------------|

| 4-6 9 146 | 7 3 2 | 145 8 a1456 |

| 34678 148 5 | 9 16 168 | 1347 134 2 |

| 3678 2 1367 | 4 156 1568 | 1379 13 1679 |

*--------------------------------------------------------------------------------*

ALS XZ Rule: X = 9, Z = 6: (6=9) r23567c9 - (9=6) r1c124789 => - 6 r7c1

This is followed by some basic moves that I assume you are familiar with :

Locked Triple {168} r8c256. Box/Row intersection 6r8c56. Locked Quad {2357} r3458c1. Locked Triple {168} in Cells r7c3 r8c2 r9c1.

This gets you to here, where a third ALS XZ Rule can be applied.

- Code: Select all
`*--------------------------------------------------------------------------------*`

|b68 b458 469 | 15 2 3 | 14789 145 14789 |

| 1 b458 b34 | 6 9 7 | 2 45-3 48 |

| 235 7 239 | 15 8 4 | 139 6 19 |

|--------------------------+--------------------------+--------------------------|

| 57 145 8 | 2 1567 156 | 145 9 3 |

| 25 6 12 | 3 4 9 | 158 7 158 |

| 9 3 147 | 8 157 15 | 6 2 145 |

|--------------------------+--------------------------+--------------------------|

| 4 9 16 | 7 3 2 | 15 8 156 |

| 37 18 5 | 9 16 168 | 347 34 2 |

|a68 2 7-3 | 4 a15 a158 | 1379 a13 1679 |

*--------------------------------------------------------------------------------*

ALS XZ Rule: X = 6, Z = 3: (3=6) r9c1568 - (6=3) r1c12, r2c23 => - 3 r2c8, r9c3.

The puzzle solves in singles from there. Hope this helps.

Leren