This puzzle from ttt demonstrates a type of opening that although probably not new, I've never been conscious of before. It exists when a house contains an ANS which can grow in steps by adding a cell at a time.

..1.7.......8..4..8.......31.3.2..8...45.37...5..1.3.22.......9..6..4.......9.1.. ttt 2009

**Preliminary Steps:**Show

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

| 3456 346 1 | 2369 7 2569 | 2689 256 568 |

| 356 367f 29 | 8 356 12569 | 4 12567 1567a |

| 8 67e 29 | 1269 4 12569 | 269 12567 3 |

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

| 1 69d 3 | 4679 2 679 | 56 8 456 |

| 69c 2 4 | 5 8 3 | 7 169 16b |

| 7 5 8 | 469 1 69 | 3 469 2 |

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

| 2 1348 57 | 1367 356 15678 | 568 34567 9 |

| 39 1389 6 | 1237 35 4 | 28 2357 578 |

| 34 348 57 | 2367 9 25678 | 1 234567 45678 |

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

In row2 cell r2c9 can be added to (3567)ANS:r2c125 to make (13567)ANS:r1259 as (1) is the only non-ANS digit in that lone cell.

The complementary AHS would then reduce from (129+x)AHS:r2c3689 to (29+x)AHS:r2c368.

The trick is to isolate the lone cell and use the smaller (3567)ANS:r2c125 with the smaller (29+x)AHS:r2c368.

If a chain from the non-ANS digit in the lone cell to any digit in the ANS can be found showing they can't be true together, then the other ANS digits can be eliminated from the smaller AHS.

(129)AHS:r2c368 = (1*)r2c9 - (1=6)r5c9 - (6)r5c1 = (6)r4c2 - (6=7)r3c2 - (7*=356)ANS:r2c125 => r2c68 <> 56

Row 2 actually provides a progression of nested ANSs and lone cell options:

(3567)r2c125 -> (1*3567)r2c1259* -> (12*3567)r2c1258*9 -> (1235679*)r2c1256*89

However the chain given seems to be the only one that produces eliminations.

A second example occurs in row 3 using (269)ANS:r3c37, (5+x)AHS:r3c68, (1269)LoneCell:r3c4

(15)r3c68 = (1*)r3c4 - (1)r8c4 = (1-8)r8c2 = (8)r8c79 - (8=56)r47c7 - (6*=29)r3c37 => r3c6 <> 29, r3c8 <> 2

In this case a segment of the external chain can be re-used to also eliminate an ANS digit from the lone cell

(2)r12c6 = (2-8)r9c6 = (8)r7c6 - (8=56)r47c7 - (6*=29)r3c37 => r3c4 <> 2

I feel that there is a tendency for solvers to ignore AHSs assuming that any eliminations they may provide will always be available through the complementary ANSs, but as these examples show, that can lead to opportunities being missed. Indeed, personally I find searching for AHSs easier.

(Although I was pleased to find these eliminations, I was disappointed to discover that I couldn't complete the puzzle using only linear methods.)

DPB

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TAGdpbNestedANSs