.

3. The Classic SK Loop 1.....9...3...7.4...5.....2.....6.7.....1.....4.3.8...9.......5.7.8...3...2...1..#8668;tax

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

| 1 . . | . . . | 9 . . |

| . 3*. | . . . | . 4*. |

| . . 5 | . . . | . . 2 |

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

| . . . | . . . | . . . |

| . . . | . . . | . . . |

| . . . | . . . | . . . |

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

| 9 . . | . . . | . . 5 |

| . 7*. | . . . | . 3*. |

| . . 2 | . . . | 1 . . |

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

Inspecting the givens for this puzzle, four of them in different boxes at r28c28 make a rectangle and are accompanied by two other diagonal givens to fill in each box (known as base cells). The four cells are potential pivot cells for the domino cell pairs in a possible loop. The two sets of digits held by the four pivot cells (347) and the other eight base

* givens (1259) should have no digits in common. If these conditions are satisfied then an SK loop will probably exist, and should be investigated further by checking that there are no more than four candidates in the cell pairs in the loop. If this is so then the linking digits in the boxes will be drawn from the base set and those in the rows and columns from the complementary digits.

* This term is used because if an Exocet pattern exists in combination with the loop, these digits will occupy its base cells.

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

| <1> 268 47 | 2456 234568 2345 | <9> 568 37 | (29)b1

(68)r2 | 268 <3*> 689 | 1259-6 259-68 <7> | 568 <4*> 168 |

| 47 689 <5> | 1469 34689 1349 | 37 168 <2> | (15)b3

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

| 2358 1259-8 1389 | 2459 2459 <6> | 23458 <7> 13489 |

| 235678 259-68 36789 | 24579 <1> 2459 | 234568 259-68 34689 |

| 2567 <4> 1679 | <3> 2579 <8> | 256 1259-6 169 |

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

| <9> 168 3468-1 | 12467 23467 1234 | 4678-2 268 <5> | (15)b7

(46)r8 | 456 <7*> 146 | <8> 259-46 1259-4 | 246 <3*> 469 |

| 3468-5 568 <2> | 45679 345679 3459 | <1> 689 4678-9 | (29)b9

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

(68)c2 (68)c8

This is the SK loop with the pivot givens asterisked. There are of 8 nodes of two cells each contained by four boxes, two rows, and two columns and their linking pairs in each house are shown alongside the grid:

(29=68)r13c2 - (68=15)r79c2 - (15=46)r8c13 - (46=29)r8c79 -

(29=68)r79c8 - (68=15)r13c8 - (15=68)r2c79 - (68=29)r2c13 - SK Loop

=> 16 Elims: 8r4c2, 68r5c2, 1r7c3, 5r9c1, 46r8c5, 4r8c6, 2r7c7, 9r9c9, 68r5c8, 6r6c8, 6r2c4, 68r2c5

Each repeating digit pair will be locked in the four pattern cells so eliminating them from the other cells in the same house.

Unusually, because of the distribution of the (6)s, there is a second way to build a loop through these cells using a 1/3 digit split:

(296=8)r13c2 - (8=156)r79c2 - (156=4)r8c13 - (4=269)r8c79 -

(296=8)r79c8 - (8=156)r13c8 - (156=8)r2c79 - (8=269)r2c13 - DLoop

=> 14 Elims: 8r45c2, 16r7c3, 56r9c1, 4r8c56, 26r7c7, 69r9c9, 8r5c8, 8r2c5

Depending on the loop used, one or other (6)4-Fish will be exposed and, once those eliminations are made, the same grid is produced.

The reduced grid contains this sub-pattern which derives an extra inference using the SK Loop.

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

| 1 . . | 5 5 5 | . 5 . |

| . . . | 15 5 . | 5 . 1 |

| . . 5 | 1 . 1 | . 1 . |

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

| 5 15 1 | 5 5 . | 5 . 1 |

| 5 5 . | 5 1 5 | 5 5 . |

| 5 . 1 | . 5 . | 5 15 1 |

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

| . 1 . | 1 . 1 | . . 5 |

| 5 . 1 | . 5 15 | . . . |

| . 5 . | 5 5 5 | 1 . . |

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

Consider the right hand pairs (15)r79c2 & (15)13c8 in the SK loop which must hold the same number of truths.

a) If they both hold no truths then (1,5)r4,5c2 & (5,1)r5,6c8 will be forced producing a contradiction for (5).

b) If they both hold two truths then (1,5)r2c4,5 & (5,1)r8c5,6 will be forced giving a second (5) contradiction.

The derived inference is that each digit pair in the loop is therefore a Single Truth Pair (STP) with a conjugate inference between the two digits.

This sub-pattern exists when the same base digits occur in diagonal boxes in the loop, digits (1&5), and one of them is a given, (1), in a cell, r5c5, where it restricts how the non-loop cells in the rows and columns can be occupied. When only one of the restrictions a) & b) operates, the derived inference will be weaker; the left hand digit pairs in each node must hold 0 or 1 truths and the right hand one must hold 1 or 2 truths or vice versa.

.