The New Sudoku Players' Forum

by **yzfwsf** » Tue Jan 18, 2022 2:33 am

Code: Select all: .......39.9...4..88.5..7......8..67.2.......4.67..1......6..3.11..2...9.95.......

Website · by **denis_berthier** » Tue Jan 18, 2022 6:39 am

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Code: Select all: +-------+-------+-------+ ! . . . ! . . . ! . 3 9 ! ! . 9 . ! . . 4 ! . . 8 ! ! 8 . 5 ! . . 7 ! . . . ! +-------+-------+-------+ ! . . . ! 8 . . ! 6 7 . ! ! 2 . . ! . . . ! . . 4 ! ! . 6 7 ! . . 1 ! . . . ! +-------+-------+-------+ ! . . . ! 6 . . ! 3 . 1 ! ! 1 . . ! 2 . . ! . 9 . ! ! 9 5 . ! . . . ! . . . ! +-------+-------+-------+

SER = 9.2

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 467 1247 124 ! 15 12568 2568 ! 12457 3 9 ! ! 367 9 123 ! 135 12356 4 ! 1257 1256 8 ! ! 8 1234 5 ! 139 12369 7 ! 124 1246 26 ! +-------------------+-------------------+-------------------+ ! 345 134 1349 ! 8 23459 2359 ! 6 7 235 ! ! 2 38 389 ! 3579 35679 3569 ! 159 15 4 ! ! 345 6 7 ! 3459 23459 1 ! 2589 258 235 ! +-------------------+-------------------+-------------------+ ! 47 2478 248 ! 6 45789 589 ! 3 2458 1 ! ! 1 3478 3468 ! 2 34578 358 ! 458 9 567 ! ! 9 5 23468 ! 1347 13478 38 ! 248 2468 267 ! +-------------------+-------------------+-------------------+ 205 candidates.

Not quite as many candidates as jovi' parquet puzzle (it had 222).

There's an obvious central symmetry, but I'll leave this to others to use.

This puzzle is interesting to me for another reason: its W rating is 19, its B rating is 13, but its gW rating is only 7.
So, this is one more example of g-whips being more useful than braids.

Simplest-first solution in gW7: Show

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by **champagne** » Tue Jan 18, 2022 7:06 am

yzfwsf wrote:
Code: Select all
.......39.9...4..88.5..7......8..67.2.......4.67..1......6..3.11..2...9.95.......

Nice shot yzfwsf. Usually the central symmetry does not give so many settings at the start with basic moves

by **DEFISE** » Tue Jan 18, 2022 1:04 pm

Symmetry / central case with following matches : 1-8 2-4 3-5 6-7 9-9

=> r5c5 = 9

Basics without using symmetry =>

Single(s): 9r3c4, 9r4c3, 1r4c2, 6r5c6, 7r5c4, 9r6c7, 8r6c8, 9r7c6
Box/Line: 3r5b4 => -3r4c1 -3r6c1
Single(s): 3r2c1, 7r2c7, 3r3c5, 6r1c1, 7r1c2, 7r7c1, 6r2c5
Box/Line: 1r3b3 => -1r1c7 -1r2c8
Box/Line: 5r5b6 => -5r4c9 -5r6c9
Single(s): 5r8c9, 5r7c5, 6r8c3, 7r8c5, 7r9c9, 6r9c8, 6r3c9
Box/Line: 8r7b7 => -8r8c2 -8r9c3
Box/Line: 2b2r1 => -2r1c3 -2r1c7
Box/Line: 4b8r9 => -4r9c3 -4r9c7
Naked pairs: 15c4r12 => -5r6c4 -1r9c4
Single(s): 5r6c1, 4r4c1, 2r4c5, 3r4c9, 5r4c6, 4r6c5, 3r6c4, 2r6c9, 4r9c4, 2r1c6, 8r1c5, 1r9c5

Code: Select all: |--------------------------------------------| | 6 7 14 | 15 8 2 | 45 3 9 | | 3 9 12 | 15 6 4 | 7 25 8 | | 8 24 5 | 9 3 7 | 124 124 6 | |------------------------------------------- | | 4 1 9 | 8 2 5 | 6 7 3 | | 2 38 38 | 7 9 6 | 15 15 4 | | 5 6 7 | 3 4 1 | 9 8 2 | |--------------------------------------------| | 7 248 248 | 6 5 9 | 3 24 1 | | 1 34 6 | 2 7 38 | 48 9 5 | | 9 5 23 | 4 1 38 | 28 6 7 | |--------------------------------------------|

whip[2]: b1n4{r1c3 r3c2}- r8n4{c2 .} => -4r1c7
STTE

by **Mauriès Robert** » Tue Jan 18, 2022 2:07 pm

Hi François,
Bravo for your resolution exploiting this central symmetry.
Do you know how to demonstrate the consequences (validations, eliminations) of such symmetries, or where I can find references for this demonstration.
Thanks to you.
Robert

by **DEFISE** » Tue Jan 18, 2022 2:31 pm

Mauriès Robert wrote:Do you know how to demonstrate the consequences (validations, eliminations) of such symmetries, or where I can find references for this demonstration.

Hi Robert,
I did not reflect on the demonstration of these properties.
Simply I am inspired by Cenoman's resolutions for similar past puzzles.
François.

by **Cenoman** » Tue Jan 18, 2022 3:41 pm

Mauriès Robert wrote:Do you know how to demonstrate the consequences (validations, eliminations) of such symmetries, or where I can find references for this demonstration.

Hi Robert,
I have not kept the link to such a demo. But make a search on this site with the key words GSP (Gurth's Symmetry Placement or Gurth's theorem.
The following thread is also very interesting: (automorphisms). Contains a deep study !

But I must give you a warning: the idea of Gurth's theorem is roughly as follows: if the givens of a puzzle present a symmetry, and if the puzzle has a unique solution, then the solution of the puzzle has the same symmetry as the givens. So, using symmetry properties to solve a puzzle is a Uniqueness Technique. If you use it, you will break a taboo

Website · by **denis_berthier** » Tue Jan 18, 2022 4:47 pm

Mauriès Robert wrote:Bravo for your resolution exploiting this central symmetry.
Do you know how to demonstrate the consequences (validations, eliminations) of such symmetries, or where I can find references for this demonstration.

There's nothing to prove.
If there's a symmetry in the data, there must be the same symmetry in the set of solutions.
If we know the solution is unique, then the symmetry must appear in this unique solution.

In the present puzzle, this obviously implies r5c5=9 and the rest is trivial (solved by e.g. a whip[2]).

by **Mauriès Robert** » Tue Jan 18, 2022 5:09 pm

Hi Cenoman, François and Denis,
In mathematics, the demonstration of a statement always requires a demonstration.
I agree with Cenoman and Denis, the uniqueness of the solution must be acquired for the demonstration of the symmetry properties to follow.
Here is the demonstration given by Mauricio. (automorphic-sudokues-t5588.html)
"If we assume uniqueness of the solution, we have the following proposition:
If A is an automorphism of the puzzle , then A is too an automorphism of its solution.
Proof: By contradiction, suppose that A is an automorphism of the puzzle and not an automorphism of its solution. If we apply the automorphism A to the solution, the puzzle remains the same, and the solution changes (as A is not an automorphism of the solution, the solution must change when we apply the automorphism to the solution). We have constructed 2 different grids that complete the puzzle, the original solution and the morphed solution, and so the original puzzle has 2 solutions, a contradiction to the uniqueness of the solution."
So I agree with Cenoman, this resolution technique is a Uniqueness Technique.
Robert

Website · by **denis_berthier** » Tue Jan 18, 2022 5:36 pm

Hi Robert
In English, the correct word is "proof", not "demonstration".

Mauricio's proof applies only for a special case of my statement. However, a similar proof applies to my general statement (to sets of solutions; when there's no uniqueness).

BTW, what the Sudoku automorphisms are precisely is totally irrelevant here.

by **Mauriès Robert** » Tue Jan 18, 2022 6:00 pm

denis_berthier wrote:Hi Robert
Mauricio's proof applies only for a special case of my statement. However, a similar proof applies to my general statement (to sets of solutions; when there's no uniqueness).

If I understand correctly, using the symmetry of the puzzle to make eliminations or validations is not a uniqueness technique according to you ?

Website · by **denis_berthier** » Tue Jan 18, 2022 6:21 pm

Mauriès Robert wrote:
denis_berthier wrote:Hi Robert
Mauricio's proof applies only for a special case of my statement. However, a similar proof applies to my general statement (to sets of solutions; when there's no uniqueness).

If I understand correctly, using the symmetry of the puzzle to make eliminations or validations is not a uniqueness technique according to you ?

No, you don't understand correctly.
There's a general theorem that doesn't require uniqueness but applies to SETS OF SOLUTIONS.
And there's its application in case the solution is unique. This obviously requires the assumption of uniqueness if you want to use it.

by **eleven** » Tue Jan 18, 2022 7:17 pm

Following your own claim, for automorphic (digit symmetric) puzzles you can always use these symmetry techniques, because all solutions must have the same automorphism (are "equivalent"). [This is, how i misunderstood your statement "there must be the same symmetry in the set of solutions".]
If it has multiple solutions, you would end up with the same problem as for non automorphic puzzles: You have to guess a number to find a solution (and a different to find another).
[Added: now again say the opposite of what i stated before] The point is, that the solutions need not be automorphic, but if you apply a symmetry, which is in the givens, to one solution, you will arrive at another solution for the same puzzle.
At the end your "general theorem" does not say more than that the givens are in all sets of solutions (trivially keeping their symmetry).

Website · by **denis_berthier** » Wed Jan 19, 2022 2:50 am

.
I can make it still more general: if you start from a set of puzzles closed under some isomorphism (the set is supposed closed..., not each puzzle), then the set of solutions is closed under this isomorphism.
Restrict the first set to one puzzle and you get my previous statement. This still doesn't provide anything very useful.*
But, further restrict the situation by assuming uniqueness of the solution, and you can apply the usual consequences.
And, as I already wrote somewhere else, this doesn't have to be restricted to permutations of digits; any iso will do.

(*) This is a special case of a very common situation, which should better be formulated in category-theoretic terms.
Here is only another example. Take a (partial-differentia) equation with a symmetry group; then the set of solutions is invariant under this symmetry group (some people tend to conclude erroneously that the solution has the same symmetry group; but no; such a conclusion requires uniqueness).
Example of the example: take a ball at the top of another ball fixed to the ground; the equations of movement of the 1st ball have rotational symmetry wrt the common axis; therefore, the set of solutions has it also; this means the ball can start moving in any direction and the movement will be the same (modulo rotation) in any direction.
.

by **DEFISE** » Wed Jan 19, 2022 8:57 am

Mauriès Robert wrote:Hi Cenoman, François and Denis,

Here is the demonstration given by Mauricio. (automorphic-sudokues-t5588.html)
"If we assume uniqueness of the solution, we have the following proposition:
If A is an automorphism of the puzzle , then A is too an automorphism of its solution.
Proof: By contradiction, suppose that A is an automorphism of the puzzle and not an automorphism of its solution. If we apply the automorphism A to the solution, the puzzle remains the same, and the solution changes (as A is not an automorphism of the solution, the solution must change when we apply the automorphism to the solution). We have constructed 2 different grids that complete the puzzle, the original solution and the morphed solution, and so the original puzzle has 2 solutions, a contradiction to the uniqueness of the solution."
So I agree with Cenoman, this resolution technique is a Uniqueness Technique.
Robert

This Mauricio’s proof is not valid, I’m afraid.
Indeed, let P be the set of initial hints of the puzzle, S a solution of the puzzle and A an automorphism of P ( => A(P) = P).
This is to prove that: (S unique & A(P) = P) => A(S) = S
Here is the summary of Mauricio’s proof :
A(P) = P & A(S) # S => there are 2 solutions : S & A(S) => S not unique => contradiction

But what proves that A(S) is a solution ?
This is precisely what we want to prove !

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