Recently I have noticed something in relation to this.

Take for example this puzzle published in the New Zealand Herald 5 Jan 2009 rated Hard.

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

| . . . | . . . | . 7 9 |

| 8 9 . | . 7 . | . . . |

| . . 7 | 6 . 4 | 3 . . |

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

| . 8 5 | . . . | . . . |

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

| . . . | . . . | 6 1 . |

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

| . . 3 | 8 . 2 | 9 . . |

| . . . | . 1 . | . 4 2 |

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

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

With some solving I arrived at the following:

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

| 2 3 6 | . . . | 4 7 9 |

| 8 9 4 | 2 7 3 | 1 . . |

| 1 5 7 | 6 9 4 | 3 2 8 |

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

| 6 8 5 | . . . | . . . |

| 3 . 1 | . . . | . . 7 |

| . . 9 | . . . | 6 1 . |

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

| . . 3 | 8 . 2 | 9 . 1 |

| 9 6 8 | 3 1 . | . 4 2 |

| 5 1 2 | . . . | . . . |

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

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

| 2 3 6 | 15 58 158 | 4 7 9 |

| 8 9 4 | 2 7 3 | 1 56 56 |

| 1 5 7 | 6 9 4 | 3 2 8 |

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

| 6 8 5 | 1479 234 179 | 2 39 34 |

| 3 24 1 | 459 2456 569 | 258 589 7 |

| 47 247 9 | 45 23458 58 | 6 1 345 |

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

| 47 47 3 | 8 56 2 | 9 56 1 |

| 9 6 8 | 3 1 57 | 57 4 2 |

| 5 1 2 | 479 46 679 | 78 368 36 |

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

Looking at box 7, the 2 remaining numbers to be placed are 4 and 7. In box 4 the remaining numbers are 2, 4 and 7. 7 can only be placed in row 6.

Considering cell r5c2, the candidates are 2 and 4. If r5c2=2 then the remaining candidates for box 4 are 4 and 7. It is then possible for the puzzle to have 2 solutions. This means that 2 cannot be a candidate for r5c2 if the puzzle has a unique solution.

Here is a paradox. Can we really claim the puzzle has a unique solution and was solved by logic alone, if we assume it does and use that to help solve it.

Has anyone else noticed this.

For the record I continued solving by logical techniques and kept in the back of my mind that r5c2 was almost certainly 4, and for those interested, here is the final solution showing that indeed r5c2 <> 2.

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

| 2 3 6 | 5 8 1 | 4 7 9 |

| 8 9 4 | 2 7 3 | 1 6 5 |

| 1 5 7 | 6 9 4 | 3 2 8 |

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

| 6 8 5 | 1 3 7 | 2 9 4 |

| 3 4 1 | 9 2 6 | 5 8 7 |

| 7 2 9 | 4 5 8 | 6 1 3 |

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

| 4 7 3 | 8 6 2 | 9 5 1 |

| 9 6 8 | 3 1 5 | 7 4 2 |

| 5 1 2 | 7 4 9 | 8 3 6 |

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