The New Sudoku Players' Forum

[Max Beran]

The Financial Times (Saturday edition) has started publishing a Sudoku puzzle. This week's is rated a diabolical.

.7.|.6.|..9
43.|..9|..5
...|1.8|37.
---+---+---
...|...|714
...|...|...
261|...|...
---+---+---
.95|2.4|...
8..|3..|.96
...|.9.|.4.

What's diabolical though is that it's a diabolical swindle - it hasn't a unique solution. You get it out to all but 10 cells and discover that they are in independent "loops" so there are multiple solutions.

[Sue De Coq]

I agree. Logic takes you so far:

Code: Select all

 1 7 2 | 5 6 3 | 4 8 9
 4 3 8 | 7 2 9 | 1 6 5
 9 5 6 | 1 4 8 | 3 7 2
-------+-------+------
 . 8 9 | 6 . 2 | 7 1 4
 . 4 . | 9 . 1 | 6 2 .
 2 6 1 | 4 . 7 | 9 5 .
-------+-------+------
 6 9 5 | 2 1 4 | 8 3 7
 8 1 4 | 3 7 5 | 2 9 6
 . 2 . | 8 9 6 | 5 4 1

whereupon guesswork leads you to four possible solutions:

Code: Select all

I.

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

II.

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

III.

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

IV.

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


[Moschopulus]

It's interesting to look at which clues could be added to the original puzzle to give a puzzle with a unique solution. This can all be done by looking at the 4 solutions.

e.g. Putting 3 or 8 in the centre square gives a puzzle with a unique solution, grids I and II, but putting a 5 there gives a puzzle with 2 solutions.

Putting a 3 in R9C1 gives a puzzle with a unique solution, grid IV.

Putting a 3 in R5C1 would pick out grid III.

Interesting to have another newspaper example with multiple solutions, like Daily Telegraph number 3.

[Sue De Coq]

I've just noticed that the original puzzle doesn't possess rotational symmetry of order 2 because it lacks an entry in r9c1. As Moschopulus says, this value would have to be a 3 in order to guarantee a unique solution. The odd thing is that, with the 3 in place, the puzzle becomes trivial - saintly rather than diabolical.

[Moschopulus]

Yes, one can only assume the clue in R9C1 was omitted.
How curious that the puzzle is then easy, not diabolical.