The New Sudoku Players' Forum

[JP_BENTZ]

As briefly explained in my previous and first post, I have investigated a deep and global structural property which seems to be closely related to the Tridagon rule developed by Denis Berthier and which I have called "3-PIPs_Par" for reasons that will become clear later on. Preliminary definitions are needed to explain it.
Let's first define "pseudo-independent" positions (or "PIPs") in a block as positions within this block which do not share any row nor any column.
There are two ways of completely dividing a block into 3 triplets of PIPs, each of these triplets being from now on denoted as "3-PIPs".
One of the two ways of dividing a block into 3-PIPs uses three "odd" 3-PIPs, while the other uses three "even" 3-PIPs. The terms "odd" and "even" as used here refer to a property of symmetry of the links that each of the 3-PIPs builds between the rows and the columns of its block. More specifically, if the positions belonging (/ not belonging) to a 3-PIPs in a block are coded by "1" (/ respectively by "0"), and if these codes are read line by line in the block they belong to, the three odd 3-PIPs have the following names and signatures : XX1 (100-010-001), ZY1 (001-100-010) and YZ1 (010-001-100), and the three even 3-PIPs have the following names and signatures : ZZ2 (001-010-100), XX2 (100-001-010), and YY2 (010-100-001).
Let's now select ANY SET of 4 blocks located in a grid at the 4 corners of a square or of a rectangle, and ANY 3-PIPs in EACH one of these 4 blocks.
As you have already guessed, the "3-PIPs_Par" property deals with the parity of the 3-PIPS selected, and simply reads as follows : IF THE NUMBER OF ODD 3-PIPS AMONG THE FOUR SELECTED 3-PIPS IS ITSELF ODD, THEN THESE FOUR 3-PIPS CANNOT CONTAIN THE SAME TRIPLET OF FIGURES. As a result, if the 12 PIPs of these four 3-PIPs initially contain the same triplet of candidates, then this configuration has no solution as it stands (i.e. without additional information).
This property may in fact be viewed as of topological nature. Indeed, when the number of odd 3-PIPs is also odd, the paths that materialise the mutual influences between the 12 PIPs along the rows and columns of the blocks have themselves an odd number of crossings. This leads to an increase in the constraints imposed by the basic Sudoku rule of uniqueness, up to the point where this rule can no longer be satisfied with 3 figures only.
Although general, this result is in fact only of moderate use insofar as the frequency of triplets of identical figures arranged as PIPs (i.e. in pseudo-independent positions) in several blocks located at the corners of a square or rectangle is not very high.
Fortunately, the notion of 3-PIPs is rich enough to be applied to pairs of figures, and therefore to much more frequent and useful configurations.
New definitions are required to explain why and how.
Let's define any set of four blocks located in a grid at the corners of a square or rectangle as a Q_Block, and let's call a Q_3-PIPs a set of four 3-PIPs, each one of which is selected in a corresponding block of a Q_Block.
If the number of odd 3-PIPs in a Q_3-PIPs is odd, then this Q_3-PIPs will itself be qualified as odd.
In fact, odd Q_3-PIPs have a very nice property : not only the paths that materialise the mutual influences between the 12 PIPs of an odd Q_3-PIPs along the rows and columns of the blocks have themselves an odd number of crossings, but these paths always contain one simple rectangular loop.
Now, simple rectangular loops act as neutral elements with regard to the parity of the number of crossings. In other words, the removal of a simple rectangular loop from the paths that materialise the mutual influences between the 12 PIPs of any Q_3-PIPs (either odd or even) may reduce the number of crossings but cannot change the parity of this number.
Therefore, every odd Q_3-PIPs can readily be transformed into an odd Q_2-PIPs by just removing its simple rectangular loop.
In other words, if an odd Q_2-PIPs is defined as an odd Q_3-PIPs, the simple rectangular loop of which has been removed, then we get the following new general rule : THE FOUR 2-PIPS OF ANY ODD Q_2-PIPS CANNOT CONTAIN THE SAME PAIR OF FIGURES.

[eleven]

Hi JP_BENTZ, welcome to the forum.

First, your first post is not lost. There is a rule (unfortunately not announced in the 'About the forum' section), that the first posts of new users are delayed for some time (3 days ?).

Second, i don't really know, what you want to show with your PIP's. Concerning tridagons there is an easy way to see, if the PIP's form one: If in the 4 boxes the PIP cells form exactly one rectangle, it is one.

You should give examples for your findings. This would it make much easier for us to understand, what you are out for (use the "code" feature to show grids or parts of them).

[JP_BENTZ]

Thank you very much, Eleven. Indeed, my 1rst post reappeared and I corrected my 2nd one accordingly. "PIPs" qualifies 2 or 3 positions WITHIN A SAME BLOCK that do not share any row nor column. When applied to 3 positions, it is called a "3-PIPS". There are 3 possible "odd 3-PIPs" et 3 possible "even 3-PIPs" in each block. When four 3-PIPs are freely chosen in 4 respective blocks of a grid at the 4 corners of a square or rectangle, the parity of the number of odd 3-PIPs determines the possibility, or not, for the four 3-PIPs to contain the same triplet of figures. As explained in my second post, this rule can be extended to pairs of figures. For instance, it is not possible to find a pair of figures to adequately replace the 8 "X" in the first scheme below, while it is possible in the second scheme, although it looks very similar to the first one at first glance.

First scheme:
0X0----0X0
000----000
X00----X00
- - - - - - -
000----000
0X0----X00
X00----0X0

Second scheme:
0X0---0X0
000---000
X00---X00
- - - - -
000---000
X00---X00
0X0---0X0

Is it any clearer ? Sorry if it is already known (although the term "TRIdagon" does not appear to me to have been chosen to apply to pairs ?).

[coloin]

Code: Select all

+---+---+---+
|.1.|.2.|...|
|...|...|...|
|2..|1..|...|
+---+---+---+
|...|...|...|
|.2.2...|...|
|1..|.1.|...|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+


is not valid

Code: Select all

+---+---+---+
|.1.|.2.|...|
|...|...|...|
|2..|1..|...|
+---+---+---+
|...|...|...|
|1..|2..|...|
|.2.|.1.|...|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+


is valid

[JP_BENTZ]

Thank you, Coloin. Sorry, I'm new. If I'm stating the obvious, please let me know. Please also let me know if this forum contains tools for producing tables such as yours. In any case, to illustrate my point, it is directly visible on the two grids that the paths of influence between the figures have an odd number of crossings in the first diagram, which is therefore invalid, and an even number of crossings in the second diagram, which is therefore valid.

[denis_berthier]

Thank you, Coloin. Sorry, I'm new. If I'm stating the obvious, please let me know. Please also let me know if this forum contains tools for producing tables such as yours. In any case, to illustrate my point, it is directly visible on the two grids that the paths of influence between the figures have an odd number of crossings in the first diagram, which is therefore invalid, and an even number of crossings in the second diagram, which is therefore valid.

I think it would be useful if you defined what you mean by "paths of influence" and "number of crossings".

[JP_BENTZ]

Thank you Denis. OK. By "paths of influence", I mean the set of ALL the rows and columns that link ALL the PIPs that are being considered, i.e. the paths along which the basic rule of uniqueness spreads in the blocks and between them. And by "number of crossings", I just mean the total number of times that these paths (or this path) cross(es) each other (or itself). For instance, in the first (invalid) scheme, there is only one complex path which crosses itself once (i.e. an odd number of times) on the third line, second column of the first block on the left. In the second scheme, there are two paths, in fact two rectangular loops, which cross each other at two different locations (i.e. an even number of times).

[eleven]

Please also let me know if this forum contains tools for producing tables such as yours.

For the "Code" feature make your table in an editor (e.g. notepad+) with a fixed width font like Courier, copy/paste it, then mark it (e.g. with the mouse) and press the "Code" button. So it will appear the same in the Code section (with fixed char width). Press "Quote" to another post to see the formatting (in [] brackets). Press "Preview" to see, if it works.

Code: Select all

+---+---+   +---+---+
|.a.|.b.|   |.a.|.b.|
|...|...|   |...|...|
|c..|d..|   |c..|d..|
+---+---+   +---+---+
|...|...|   |...|...|
|.e.|f..|   |e..|f..|
|g..|.h.|   |.g.|.h.|
+---+---+   +---+---+


So what you mean is, that in the first sample only ae crosses cd (odd), while in the second ag crosses both cd and ef (even) ?

[JP_BENTZ]

Thank you very much for your explanations, Eleven. Regarding odd and even Q_2-PIPs, your understanding is absolutely correct. The second and valid scheme above is indeed an even Q_2-PIPs having two paths of influence, i.e. abhga and cdfec, and two crossings (an even number), i.e. ga/ef and ga/cd. In contrast, the first scheme is an odd Q_2-PIPs having a single path of influence (which is not relevant in itself), i.e. abhgcdfea, having only one (the relevant odd number) crossing, in fact "self-crossing" in ea/cd, as you correctly mentioned. Some odd Q_2-PIPs generate more complex influence paths having three self-crossings, but the main consequence (invalidity) is the same. For instance, the following invalid scheme is an odd Q_2-PIPs whose single path of influence abcdefgha has 3 self-crossings, i.e. ef/bc, gh/bc and gh/de.

Code: Select all

 +---+---+
 |a..|b..|
 |.e.|.f.|
 |...|...|
 +---+---+
 |h..|.g.|
 |...|...|
 |.d.|c..|
 +---+---+

[eleven]

Ok, as a manual solver i some "(5-cell bivalue) oddagons" in the 1st and 3rd sample, each being obviously impossible for 2 digits: abhfe/abhgc/abdfe/cdbhg and abcdh/abfgh/abcgh/abcde/efgcd/efghd (if you start with any digit, the last cell would get the same, though it "sees" the first one).
But nice to see another method to show, that they are invalid.

[JP_BENTZ]

Thank you, Eleven. I can easily agree with you on this : knowing that a specific odd Q_2-PIPs in front of us cannot contain the same pair of figures is no big deal. However, the property in question applies to each of the 110 odd Q_2-PIPs (if I am correct - rotations excluded) contained in each of the 9 Q_Blocks of a grid. It is therefore a "systemic" property likely to have other implicit though useful manifestations. It is even the case with the even Q_3-PIPs, specially the 117 ones (90+27 if I am correct - rotations excluded) contained in each Q_Block and linking all the 12 PIPs in a single path. Indeed, although any triplet of numbers can satisfy them, these numbers must be used in circular permutation, and any local inversion of their order along the path leads to an error. I'm going to continue looking into the matter.