The diagram below shows the position after initial eliminations from basic techniques, a naked triple of "189" in r8c238, a hidden pair of "14" in r1c6|r2c4, and a naked triple of "289" in r247c5:

Code: Select all

```
.-------------------.-------------------.-------------------.
| 389 2 3489 | 6 5 14 | 89 7 189 |
| 7 *69 46-9 | 14 *29 8 | 3 5 126-9 |
| 1 5 689 | 239 7 239 | 689 4 2689 |
:-------------------+-------------------+-------------------:
|*89 *689 56-89 | 125-89 *289 7 | 16-89 3 4 |
| 2 7 35689 | 13589 4 139 | 1689 89 689 |
| 389 4 1 | 389 36 369 | 7 2 5 |
:-------------------+-------------------+-------------------:
| 6 189 2 | 7 89 5 | 4 189 3 |
| 4 189 89 | 23 36 236 | 5 189 7 |
| 5 3 7 | 489 1 49 | 2 6 89 |
'-------------------'-------------------'-------------------'
```

I also did not need to make use of the threatened "36" deadly pattern in r68c56.

In the diagram, the cells marked with "*" (r2c25, r4c125) form what I might call a simple ALS ring (a generalization of an XY ring), although I'm open to suggestions if there's a standard or better name.

(6=9)r2c2 - (9=2)r2c5 - (2=896)r4c125 - (6=9)r2c2

I have come across such ring structures occasionally in Nightmare solutions but have never commented on them before. The idea is the same as in an XY ring -- the common value(s) appearing in the two nodes forming any side of the ring can be removed from any cell which sees both nodes. This means that "9" can be removed from r2c39, and that 8 and 9 can be removed from r4c347. If there were any other "6" candidates in column 2, or "2" candidates in column 5, they could be removed also.

After these eliminations, we have the position below, in which there are two deductions from uniqueness arguments:

Code: Select all

```
.----------------.----------------.------------------.
| 389 2 3489 | 6 5 14 | 89 7 189 |
| 7 69 46 | 14 29 8 | 3 5 126 |
| 1 5 689 | 239 7 239 | 689 4 2689|
:----------------+----------------+------------------:
| 89 689 *56 |*15+2 289 7 |*1-6 3 4 |
| 2 7 *56+3 |*15+3 4 13 |*16+89 89 689 |
| 389 4 1 | 389 36 369 | 7 2 5 |
:----------------+----------------+------------------:
| 6 #189 2 | 7 89 5 | 4 #189 3 |
| 4 #189 &89 | 23 36 236 | 5 #1-89 7 |
| 5 3 7 | 489 1 49 | 2 6 &89 |
'----------------'----------------'------------------'
```

(8=9)r9c9 - (9=18)r78c8 - UR - (18=9)r78c2 - (9=8)r8c3 => r8c8 <> 8

The pattern is symmetric with respect to 8 and 9 so a similar argument gives r8c8 <> 9.

The second elimination is based on the threatened 56-61-15 deadly pattern in r45c374 (marked with "*"). There is a way to avoid the deadly pattern in each of boxes 4, 5, and 6, but all escape routes lead to a common deduction, namely r4c7=1.

In box 4, digit "5" is locked into the pattern, so the only way to avoid the pair of "56" cells is to place "6" in r4c2.

(6)r4c2 => (1)r4c7.

In box 5, "5" is again locked into the pattern, so the only way to avoid the pair of "15" cells is to place "1" in r5c6.

(1)r5c6 - (1)r5c7 = (1)r4c7

In box 6, "1" is locked into the pattern, so the only way to avoid the pair of "16" cells is to place "6" in r5c9.

(6)r5c9 => (1)r4c7.

Following up these deductions, we have the following position. Here there is a threatened 12 cell deadly pattern with base candidates 8 and 9, marked with "*" in the diagram. There is a surplus candidate in six of the 12 cells, but all lead to a common deduction, namely that r2c4=1.

Code: Select all

```
.----------------.-----------------.-----------------.
| 389 2 3489| 6 5 14 |*89 7 *89+1|
| 7 69 46 | 1-4 29 8 | 3 5 126 |
| 1 5 689 | 239 7 239 | 689 4 2689|
:----------------+-----------------+-----------------:
|*89 689 56 | 25 *89+2 7 | 1 3 4 |
| 2 7 35 | 135 4 13 |*89+6 *89 689 |
|*89+3 4 1 |*89+3 36 369 | 7 2 5 |
:----------------+-----------------+-----------------:
| 6 1 2 | 7 *89 5 | 4 *89 3 |
| 4 89 89 | 23 36 236 | 5 1 7 |
| 5 3 7 |*89+4 1 49 | 2 6 *89 |
'----------------'-----------------'-----------------'
```

((8or9)=6)r5c7 - (6=891)r1c79|r3c7 - (1=4)r1c6 - (4=1)r2c4

((8or9)=4)r9c4 - (4=1)r2c4

((8or9)=2)r4c5 - (2=9)r2c5 - (9=6)r2c2 - (6=4)r2c3 - (4=1)r2c4

((8or9)=3)r6c4 - (3=1)r5c6 - (1=4)r1c6 - (4=1)r2c4

((8or9)=3)r6c1 - (3=1894)r1c1679 - (4=1)r2c4

After placing r2c4=1 the rest is straightforward.

An XY chain which gives this same conclusion is:

(4=1)r1c6 - (1=3)r5c6 - (3=5)r5c3 - (5=6)r4c3 - (6=4)r2c3 => r2c4 <> 4.