## To solve hardest Sudoku

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

### To solve hardest Sudoku

Hi, Some grids are to complex and if you don't know sophistical method you cannot solve them

This kind of problem are reserved for those wich use computer or are famillar with math. But the standard playeur, with just his pencil and his grid
never find the solution

I'm in the second position: usin computer to solve is not my trip and I 'm a b&d (vcery bad?) mathematitcen.

So how to solve this grids?
Using a non used specification of the grid: the digits renumber. Not clues only digits

We know taht on a solve grid we can remaps digits to have one set to a particular value:
The speed and simple way is to search the row,column or box with the max clues

For example it's the box 5 with 5 clues

x 9 x
1 x 6
7 2 x

We choose the specific value 123456789
The goal is to obtains a box 5 like this
1 2 3
4 5 6
7 8 9

Hox to do?
To begin
On the GRID remplace all 9 by 2 to oibtain x 2 x 1 x 3 7 9 x
Swap 1 et 4 x 2 x 4 x 6 7 9 x
Swap the 9 and 8
x 2 x
4 x 6
7 8 x

At this time no change for the solution. just swaps digits

x x x x 5 x xxx
x x x x x x xxx
x x x x x x xxx

1 x x x 2 x xxx
9 x x 4 x 6 xxx
x x x 7 8 x xx9

x x x x x x xxx
x x x x x x xxx
x x x x x x xxx

Now what numbers are missing? 1,3,5,9. What can we say about the 1 in r4,c4 like we hope? We cannot put a 1 in r4,c4 because the one in r4c1!!!
so we swap the 1 and the 9

9 x x x 2 x x x x
1 x x 4 x 6 x x x
x x x 7 8 x x x 1
So the 1 is possible IN r4,c4

We can do he same with the 5 in r1c5
Swap 5 and 9

x x x x 9 x xxx
x x x x x x xxx
x x x x x x xxx

5 x x x 2 x xxx
1 x x 4 x 6 xxx
x x x 7 8 x xx1

x x x x x x xxx
x x x x x x xxx
x x x x x x xxx

When this job is terminated you can suppose that the BOX 5 will be 123456789. his is your beginning hypothése

you can begin with this grids

x x x x 9 x xxx
x x x x x x xxx
x x x x x x xxx

5 x x 1 2 3 xxx
1 x x 4 5 6 xxx
x x x 7 8 9 xx1

x x x x x x xxx
x x x x x x xxx
x x x x x x xxx

If you arrive to impossible it's because yoiu have not well move the digits NOT in the clues

You can apply this method for Row or Column or Box
At the beginning Iit's easier to wotk on the row 1 (swapping blocs or rows)
For me this method is easy to use on paper grid.
his method is not universal but if, solving the problem, you are stopped you can use the method again changing the set of reference...

I hope that this method will help you. For me I used it, of course, and i help me...

I also change the French Paires....

Papy
Papy

Posts: 131
Joined: 15 August 2006

Why dont you assume that block 5 is
Code: Select all
`x 9 x    3 9 4      4 9 3         8 9 51 x 6    1 5 6  or  1 5 6 or .... 1 4 67 2 x    7 2 8      7 2 8         7 3 8`
or whatever is possible from the candidate grid ?
Wouldn't that save you the time for swapping the digits ?
ravel

Posts: 998
Joined: 21 February 2006

### Block5

My example for this block 5 is
x x x x 9 x x x x
x x x x x x x x x
x x x x x x x x x
5 x x x 2 x x x x
1 x x 4 x 6 x x x
x x x 7 8 x x x 1
x x x x x x x x x
x x x x x 9 x 1 x
x x x x x x x x x

There is only 9 clues!!!
Take a grid with 20 clues and you will be more efficiant.
For example the 1 can be only c4 or c7 Perhaaps an 1 on the grids can help you I set r8c6 and r8c7

If yo swap 1 and 9 The 1 cannot be in R4 C6 only in r3C4
You have to choose the set where constraints are hrdest.
Of course this way cannot give you the 9 number but changiung the challenge of solving the the grids in set 123456789 wuth all the clues
give you a strong probability to ge this particular value: Not clues on the grids must be in opposition with 123456789. Is(t more easy to do taht to solve the 81 cells.
It's juste a sample and I say: that is tour hypothese to begin
On some grids you don't now haow to begin!
Not the universal solution only an other way tfor searching .

Papy
Papy

Posts: 131
Joined: 15 August 2006

Sorry, but as i see it, you are reinventing the wheel in a more complicated form, as we know it. So please dont expect another answer by me, before i changed my mind.
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:you are reinventing the wheel in a more complicated form...

Sorry, I second that.
Mauricio

Posts: 1174
Joined: 22 March 2006

I'm perhaps 'reinventing the wheel'
But I never found this method in any message about the sudoku

In no site I find this: perhaps I don't visite your.
so my tread was for those wich are like me.

But it's very strange there is a bug in my demonstration
You don't see it? You seems to well know it rtherefore!

So please Ravel and Mauriccio where can I found this method
You have answer quickly to my theory please be so speed the indicate the well site.

One thing : is this method valid?

Papy
Papy

Posts: 131
Joined: 15 August 2006

Papy wrote:So please Ravel and Mauriccio where can I found this method
IMHO this method is the same as guessing all missing numbers in a box at once (the relabeling does not give you any more information). I saw something similar by StrmChkr. Of course guessing is valid, but not very popular.
ravel

Posts: 998
Joined: 21 February 2006

Sorry I don't know what means IMHO perhaps AMHA in french?

But It seems that you don't have understood well all he avantages and the NEWS information that you can get using it.

The first part is a sample relabelisation and you are right no effect (I recise that in my message)
You have to wait the second part to get information

1° Renumber the digits IN A SET: no thinghs more!
2° Renumber all digits that make a digit impossible in a specific cell
The interest of this method which is not universal is that you change the goal of the game.
You don't try to solve 81 cells in an unknow order: you rearange ALL the clues to put a specific value in a given set. It's more easy!
Papy
Papy

Posts: 131
Joined: 15 August 2006

IMHO = In My Humble Opinion

Papy, ravel is right. Your technique is nothing more than guessing all unsolved digits in a box. Relabeling the digits does not make your guess any more correct than any other guess without relabeling. The technique is just as effective without relabeling, only guessing whatever possible values in the empty cells.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Papy, you are only renumbering the grid, perhaps over and over.

There is no new information from a renumber.
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

Hi,

here is a sample. Isn't a very hard grid but it's difficult (for me) to begin
Using my method I find the solution it whithout XWing, triplet or other...
In French we call it Flower

3 0 0 0 7 0 0 0 8
0 2 0 0 0 0 0 9 0
0 0 1 0 0 0 4 0 0
0 0 0 1 0 6 0 0 0
8 0 0 0 5 0 0 0 6
0 0 0 2 0 8 0 0 0
0 0 4 0 0 0 5 0 0
0 9 0 0 0 0 0 1 0
6 0 0 0 8 0 0 0 7

How to solve without computer?

The goal is nort to say I find a new method , but do you know this method and and is it a valid method. Sorry if I have said it.

Papy
Papy

Posts: 131
Joined: 15 August 2006

If you somehow relabel that puzzle to get a ordered 1-9 box in the middle and then solve the rest with singles, then your method is definitely not valid. Filling in the correct missing values in box 5 leaves a puzzle with ER 8.5.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Perhaps I don't explain me well sorry

I renumber the first row and I solve the grids with
1- single digit(for mle this number can only be here in a particular set)
2- single clue(this clue can only be set to X)
3- This nmber can only be in that subset.

If at one time you ar stopped change the set and try again.
How much do you obtains for the orininal puzzle? 8.5 ore more?

Trying this method on all setsone by one give you the same ER?
Thanks
Papy
Papy

Posts: 131
Joined: 15 August 2006

Here is my solution

Renumber the grid like ths

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

r1c2 2 r1c3 3 r1c4 4 r1c6 6 r1c7 7 r1c8 8
Strating hypothése
r4c9 8 r5c3 8 r8c1 8 r9c8 8 r3c5 8
r2c1 6 r5c4 5
r5c6 4 r7c5 4 r9c7 4 r6c9 4 rcc1 4
r7c6 2 r9c3 2 r6c1 2 r2c5 2 r3c9 2 r5c7 2
r8c6 5 r2c9 3 r7c9 7 r8c9 6 r7c1 5 r3c1 7
r3c4 3 r8c5 3 r7c8 3 r6c7 3 r5c2 3 r9c4 6
r8c3 7 r2c4 7 r4c2 7 r5c8 7 r6c2 5 r4c8 6
r3c2 5 because if you set r2c3 to 5 you cannot put a 5 in box 3!
r2c7 5 and it's ended juste one coice by cell

So for me I have only use elementary methods

It's the same tjhat canonize a grid to give it always the same 1 row ot column or box.

I never solve this grid otherway.
Papy
Papy

Posts: 131
Joined: 15 August 2006

Papy wrote:Renumber the grid like ths

Papy, that is not the same puzzle as the one you started with. Your method of relabeling is not valid. If you relabel digits, then you must relabel all the same digits the same way. Otherwise you end up with a different puzzle, as in this case. You still haven't solved the original puzzle.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Next