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 mcarne5252
 Joined: 20 Apr 2006  Posts: 1  :   Items 

Posted: Thu Apr 20, 2006 1:51 am Post subject: Unique Solution 


Can someone provide an explination or proof, perhaps using the idea of Latin Squares, on why a proper sudoku puzzle will always have one unique solution?
Thanks 

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 Ruud Site Admin
 Joined: 17 Sep 2005  Posts: 708  :  Location: Netherlands  Items 

Posted: Thu Apr 20, 2006 2:13 am Post subject: 


I'm not sure what you're asking for.
A sudoku is a proper sudoku because it has a unique solution. It is a qualification by convention.
You can examine a sudoku with a backtracking solver. When the solver finds exactly one solution, you may label it 'proper'.
If this is not what you're looking for, please explain what you need to know,
Ruud. _________________ Meet me at sudocue.net 

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 coloin
 Joined: 05 May 2005  Posts: 97  :   Items 

Posted: Thu Apr 20, 2006 8:38 am Post subject: 


There are three aspects here:
the final grid
the given clues
the generated clues
the final grid has to be valid before you start
Code:  ++++ Complete grid
439286157
176593284
258147396
++++
381754629
795632841
624918735
++++
542371968
867429513
913865472
++++ 
all variations of the given glues [taken from the vaild completed grid] will therefore be valid
the question of whether it will be solved uniquely is down to whether each "unavoidable set" in the grid has at least one given clue in it.
Each complete grid has THOUSANDS of unavoidable sets !  it is remarkable that some grids can be solved in 17 clues....but almost all grids have a combination of 19 given clues which cover all the sets and define the grid.
The simplest unavoidable sets have 4 clues  one of these clues in the set must be one of the given clues.
Code:  ++++4 set unavoidable  this has 2 grid solutions
439286157
176593284
258147396
++++
381754629
795632841
6..918735
++++
5..371968
867429513
913865472
++++
++++ 6 set unavoidable  2 solutions
439.86.57
.76593.84
.58.47396
++++
381754629
795632841
624918735
++++
542371968
867429513
913865472
++++ 
there are also 36[8+7+6+5+4+3+2+1],18 clue unavoidable sets consisting of 9 of one number and 9 of another.......this explains why you can only have one absent clue number.
Code:  ++++18 set unavoidable  no3s and no4s!  this has 2 grid solutions
..9286157
17659.28.
2581.7.96
++++
.8175.629
7956.28.1
62.9187.5
++++
5.2.71968
867.2951.
91.865.72
++++ 
a generated clue can be inserted when ALL the unavoidable sets with this clue in it already have in them at least one given clue.
the program "unavoid" in "checker" will print out all the smaller unavoidable sets in any given grid. http://www.maths.nuim.ie/staff/gmg/sudoku/
I hope this explains to all !
Last edited by coloin on Thu Apr 20, 2006 3:34 pm; edited 1 time in total 

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 Papy
 Joined: 14 Jul 2006  Posts: 12  :   Items 

Posted: Thu Aug 17, 2006 11:13 am Post subject: Unique solutionn 


You don't have to proof that there is only one solution:
by definition a sudoku grid have only one solution!!!
Multiply solutions don't dive sudoku
Papy 

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 Catweazle
 Joined: 30 Jul 2006  Posts: 8  :  Location: Brazil  Items 

Posted: Fri Aug 18, 2006 2:42 am Post subject: 


I agree with Papy.
If a sudoku has more than one solution, you simply cannot solve the puzzle with logic.
And therefore it is no puzzle because you cannot finish it.
This because at a certain point you do not have a clue which number to fill in next.
Therefore a sudoku puzzle has to have 1 solution so it can be solved with logic until the end.
I hope this answers your question fully.
Theo _________________ More puzzles please...... 

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