- Code: Select all
`Grid:`

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

|.6.|154|928|

|425|968|731|

|...|732|654|

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

|..2|641|873|

|...|579|162|

|176|283|495|

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

|...|316|289|

|2..|897|546|

|689|425|317|

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

Candidates:

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

| 37 6 37 | 1 5 4 | 9 2 8 |

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

| 89 19 18 | 7 3 2 | 6 5 4 |

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

| 59 59 2 | 6 4 1 | 8 7 3 |

| 38 34 348 | 5 7 9 | 1 6 2 |

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

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

| 57 45 47 | 3 1 6 | 2 8 9 |

| 2 13 13 | 8 9 7 | 5 4 6 |

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

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

and if you drop this into Simple Sudoku and ask for a hint, it will tell you "No hint available". This is the smallest configuration (15 open cells) I have yet encountered for which elementary methods (i.e. the methods known to Simple Sudoku) are of no avail.

It has some interesting properties: there is only one "abundant" digit, namely 3, and one "abundant" cell, namely r5c3. If it were possible to remove the 3 as a candidate from r5c3, then the remaining cells would either have no solution or more than one solution. The easiest way to see this is to note that if the reduced candidate matrix has a solution, then there must be a second solution created by replacing each digit selected in the open cells by its alternative. Thus, if there is a unique solution, r5c3 must be a 3, which is indeed the case.

I'm pretty sure that there are several other ways of solving this grid. Can anyone suggest one which is simpler (preferably not relying on uniqueness arguments)?