Paradoxical Solving technique

Everything about Sudoku that doesn't fit in one of the other sections

Paradoxical Solving technique

Postby Condor » Wed Mar 11, 2009 3:21 am

It was argued right from the beginning, a sudoku puzzle must have 1 solution to be a valid puzzle. Indeed if it is solved by logic alone it must 1 solution.

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 |
 *-----------------------*
Condor
 
Posts: 62
Joined: 19 June 2005

Postby daj95376 » Wed Mar 11, 2009 4:11 am

The pro and con discussions on Uniqueness Testing are long and bloody. I doubt if many are interested in resurrecting them.

You might wish to check the information in Sudopedia on this topic.
daj95376
2014 Supporter
 
Posts: 2624
Joined: 15 May 2006

Postby StrmCkr » Wed Mar 11, 2009 9:16 am

the ur may be the easier way to solve it but yes it still leaves the open and heavily debated argument is it still only have 1 solution. as a ur can remove that as an option.

there is other moves but many are much more complex and will still arive at the same solution depiced above.

it mostly comes down to a choice of the solver to trust the creater left only 1 soltution and use these type of moves.

or try to prove it is with out a doubt unique, and not use them.
Some do, some teach, the rest look it up.
User avatar
StrmCkr
 
Posts: 647
Joined: 05 September 2006

re: assuming Uniqueness-Of-Answer

Postby Pat » Wed Mar 11, 2009 7:22 pm

Condor wrote:Here is a paradox:
Can we really claim the puzzle has a unique solution,
if we assume it does and use that to help solve it?

no, if we assumed Uniqueness-Of-Answer and used that to reach an answer, this does not prove that the puzzle has only one answer.
    but i suspect you knew that.
    no paradox.
User avatar
Pat
 
Posts: 3438
Joined: 18 July 2005

Postby Condor » Sun Mar 15, 2009 6:34 am

daj95376 wrote:The pro and con discussions on Uniqueness Testing are long and bloody. I doubt if many are interested in resurrecting them.


I was not trying to resurrect a discussion on whether puzzles are valid if they can be solved by logic alone or if they can only have 1 solution.

What a person thinks about puzzles having unique solutions was not the point of my posting but whether you can claim the puzzle has a unique solution if you start with the assumtion the puzzle has a unique solution.
Condor
 
Posts: 62
Joined: 19 June 2005

Postby StrmCkr » Sun Mar 15, 2009 6:38 am

then that would be the folly of the solver,

to make such a claim while trying to prove it is infact unique.

while utilizing moves that full well remove solutions counts.

thats not a paradox either.
Some do, some teach, the rest look it up.
User avatar
StrmCkr
 
Posts: 647
Joined: 05 September 2006

Postby Smythe Dakota » Sun Mar 15, 2009 9:12 am

If you reject a candidate for a cell because that candidate would result in multiple solutions, all you have proved is that if the solution is unique, then that candidate can be rejected.

Some people use this technique to arrive at an answer more quickly than they could otherwise. They use it on puzzles (such as newspaper puzzles) that have a solid reputation of being "proper" puzzles, i.e. of having only one solution.

No paradox.

I might add that I have recently seen Kakuro puzzles, also, which lend themselves to this technique. I personally don't like the technique, either in Kakuro or Sudoku, as it offends my purist genes. But, to each his own.

Bill Smythe
Smythe Dakota
 
Posts: 534
Joined: 11 February 2006


Return to General