biG wrote:Playing with Grids - From First to Last Still Working on this - July 2, 2025
Definitions
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Grids
I copied the grids from my RefPOST (refer to my Defintions) into a text editor and edited them to create these grids.
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c1 = 1 2 3 4 5 6 7 8 9 c1 = 1 2 3 4 5 6 7 8 9
.---------.---------.---------. .---------.---------.---------.
r1 | 1 2 3 | 4 5 6 | 7 8 9 | r1 | 9 8 7 | 6 5 4 | 3 2 1 |
r2 | 4 5 6 | 7 8 9 | 1 2 3 | r2 | 6 5 4 | 3 2 1 | 9 8 7 |
r3 | 7 8 9 | 1 2 3 | 4 5 6 | r3 | 3 2 1 | 9 8 7 | 6 5 4 |
:---------+---------+---------: :---------+---------+---------:
r4 | 2 1 4 | 3 6 5 | 8 9 7 | r4 | 8 9 6 | 7 4 5 | 2 1 3 |
r5 | 3 6 5 | 8 9 7 | 2 1 4 | r5 | 7 4 5 | 2 1 3 | 8 9 6 |
r6 | 8 9 7 | 2 1 4 | 3 6 5 | r6 | 2 1 3 | 8 9 6 | 7 4 5 |
:---------+---------+---------: :---------+---------+---------:
r7 | 5 3 1 | 6 4 2 | 9 7 8 | r7 | 5 7 9 | 4 6 8 | 1 3 2 |
r8 | 6 4 2 | 9 7 8 | 5 3 1 | r8.| 4 6 8 | 1 3 2 | 5 7 9 |
r9 | 9 7 8 | 5 3 1 | 6 4 2 | r9 | 1 3 2 | 5 7 9 | 4 6 8 |
'---------'---------'---------' '---------'---------'---------'
A114288 (first: minlex) A112454 (last: maxlex)
c1 = 1 2 3 4 5 6 7 8 9 c1 = 1 2 3 4 5 6 7 8 9
.---------.---------.---------. .---------.---------.---------.
r1 | 2 3 | 5 6 | | r1 | | | |
r2 | 4 5 6 | 7 8 9 | | r2 | B1 | B2 | B3 |
r3 | 7 8 9 | 1 2 3 | | r3 | | | |
:---------+---------+---------: :---------+---------+---------:
r4 | 1 4 | | 8 9 7 | r4 | | | |
r5 | 3 6 5 | | 2 1 4 | r5 | B4 | B5 | B6 |
r6 | 8 9 7 | | 3 6 | r6 | | | |
:---------+---------+---------: :---------+---------+---------:
r7 | | 6 4 2 | 9 7 8 | r7 | | | |
r8 | | 9 7 8 | 5 3 1 | r8.| B7 | B8 | B9 |
r9 | | 5 3 | 6 4 | r9 | | | |
'---------'---------'---------' '---------'---------'---------'
"The 48" (33 holes) Block Numbering
c1 = 1 2 3 4 5 6 7 8 9 c1 = 1 2 3 4 5 6 7 8 9
.---------.---------.---------. .---------.---------.---------.
r1 | 1 2 3 | 4 5 6 | | r1 | . . . | . . . | . . . |
r2 | 4 5 6 | 7 8 9 | | r2 | . . . | . . . | . . . |
r3 | 7 8 9 | 1 2 3 | | r3 | . . . | . . . | . . . |
:---------+---------+---------: :---------+---------+---------:
r4 | 2 1 4 | | 8 9 7 | r4 | . . . | . . . | . . . |
r5 | 3 6 5 | | 2 1 4 | r5 | . . . | . , . | . . . |
r6 | 8 9 7 | | 3 6 5 | r6 | . . . | . . . | . . . |
:---------+---------+---------: :---------+---------+---------:
r7 | | 6 4 2 | 9 7 8 | r7 | . . . | . . . | . . . |
r8 | | 9 7 8 | 5 3 1 | r8.| . , . | . . . | . . . |
r9 | | 5 3 1 | 6 4 2 | r9 | . . . | . . . | . . . |
'---------'---------'---------' '---------'---------'---------'
"The 54" (27 holes) Next?
My idea was to demonstrate the 48-char compression with a specific example. Start with the first minlex grid in the ED catalog Since we are documenting, lets draw the last grid (the maxlex one in the ED catalog. Observe as we go.
The first thing I notice is that the first grid A114288 could be transformed into that last grid A112454 in a modulo-10 sense. E.G. iterating into the values of the fisrt grid: subtract the value from 10 and that was the corresponding values in the last grid. Made me go "hmmm..." Seems like there is a symmetry here. Could I just "calculate" the first ED/2# grids and generate the last half by applying modulo-10 operations. My gut - feeling says the last half of the ED is geneated from the end to the middle. What say you?
Then in the process of decompressing The 48 grid by doing a "full house" on six blocks produces The 54" grid. Of course by experience I "know" the grid can be solved from there to the full 81-char puzzle. However, I was still wanting to see this visually in some way. It made me think of that so-called SET (Set Equivalence Theory) method again! Where do I go now?
****************************************************** Continuing - July 2, 2025
The principle from The 54 grid is sometines called the pigeon hole principle. Similar to Full House or Naked Single. I was having trouble visualizing this so I have decided to play this in HoDoKu for this first minlex grid
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0 1 2 3 4 5 6 7 8
123456789012345678901234567890123456789012345678901234567890123456789012345678901
---------------------------------------------------------------------------------
v
.23.56...456789...789123....14...897365...214897...36....642978...978531...53.64.
235645678978912314897365214897366429789785315364.
.---------.---------.---------.
| 1 2 3 | 4 5 6 | . . . |
| 4 5 6 | 7 8 9 | . . . |
| 7 8 9 | 1 2 3 | . . . |
:---------+---------+---------:
| 2 1 4 | . . . | 8 9 7 |
| 3 6 5 | . . . | 2 1 4 |
| 8 9 7 | . . . | 3 6 5 |
:---------+---------+---------:
| . . . | 6 4 2 | 9 7 8 |
| . . . | 9 7 8 | 5 3 1 |
| . . . | 5 3 1 | 6 4 2 |
'---------'---------'---------'
HoDoKu Candidates Diagram
The candies (holes) are the values in round brackets (I added)
.---------.---------.---------.
| 1 2 3 | 4 5 6 |(7)(8)(9)|
| 4 5 6 | 7 8 9 |(1)(2)(3)|
| 7 8 9 | 1 2 3 |(4)(5)(6)|
:---------+---------+---------:
| 2 1 4 |(3)(6)(5)| 8 9 7 |
| 3 6 5 |(8)(9)(7)| 2 1 4 |
| 8 9 7 |(2)(1)(4)| 3 6 5 |
:---------+---------+---------:
|(5)(3)(1)| 6 4 2 | 9 7 8 |
|(6)(4)(2)| 9 7 8 | 5 3 1 |
|(9)(7)(8_| 5 3 1 | 6 4 2 |
'---------'---------'---------'
From this specific [minlex] diagram it is obvious that just exerting the singles solves this back to a full 81-char solution. Now a good argument from here would be if that this specific puzzle is solved then by ANY consistent mapping of the digits to any other digits would work as well! This would be true all possible proper puzzles.
Now, just for fun I will attempt to make another argument from the following image. Open the following link in a separate tab:
https://drive.google.com/file/d/1aM2ngoSCeS9Gd4iBXVY0jzXON_c4goEw/view
Then follow along in this reasoning. We are ONLY concerned with occuppancy in other areas, not their specific order. Let's start out trying "generally" solving for the square colored in RED. The first 6 digits in row 1 are KNOWN. These digits MUST occupy the cells highlighted in liight-orange. Further more the last two digits in row 1 highlghted in purple Can NOT be in the light-orange region.
Now consider c7 except for the red cell we are solving for. There are 8 cells here. We know they must contain just by occupancy requirement the 6 orange values which are ALL KNOWN. In addition the puple digits (whatever they are) must be here also, which makes them now known. The r23c7 digits are KNOWN and must be from the orange group. Thus 8 of the values in c7 are previously known. Then by the Full House / Naked Single principle the red square is the only single left.
That general argument can be continued for all the other blocks and their values. It helped me see this generally and infer that the 48-char grid is sufficient to encode / decode any proper Sudoku puzzle.
