3

Definitions

For a $9\times9$ grid $G$ (not necessarily respecting sudoku rules), a subset of cells $S$ is defined to be sudoku-friendly if the values in $S$ do not contradict the rules of sudoku.
This means that in $S$ no number occurs twice in the same row, column or box.

Take the grid $G$:

111111111
222222222
333333333
444444444
555555555
666666666
777777777
888888888
999999999

Then the following is an example of a sudoku-friendly subset $S$ of $G$:

1        
 2
  3
   4
    5
     6
      7
       8
        9

A net is a mapping on a grid that produces a subset of cells. You can imagine it like a piece of paper with holes in it, lying on the grid, which hides some cells.
This is usually visualized with O (visible) and X (blocked) cells.

A net has a visible set, $V_N(G)$ which is defined as the set of cells in a grid $G$ which map to an O in $N$, namely the set of visible cells.

The net which retrieves the above visible set $S$ would look like this:

OXXXXXXXX
XOXXXXXXX
XXOXXXXXX
XXXOXXXXX
XXXXOXXXX
XXXXXOXXX
XXXXXXOXX
XXXXXXXOX
XXXXXXXXO

A sudoku net is a net whose visible set is sudoku-friendly.

Challenge

Construct a sudoku net $N$ of maximum and minimum possible blocks such that there exists some 9x9 $G$ that is an impossible sudoku puzzle (namely no solution).

-Bounty awarding question:

  • is there a grid which has unique solution whatever are the arrangements of legal patterns in friendly-sudoku ?
Gareth McCaughan
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Abr001am
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  • Please describe how this maximal net differs from the maximal net in the linked question. – Ian MacDonald Apr 21 '15 at 20:32
  • i can still roll it back to first non edited version if it is duplicate – Abr001am Apr 21 '15 at 20:36
  • @Ian MacDonald the solution shown in previous puzzle doesnt fit my puzzle – Abr001am Apr 21 '15 at 20:37
  • the bold line is the difference – Abr001am Apr 21 '15 at 20:40
  • Your wording is still unclear. You want a net that produces a unique solution, but you start with an empty grid. For any sudoku grid, there are 9 identically structured solutions that can be reached by cycling the digits. You need to explicitly define how a solution is "unique" when reached from an empty netted grid. – Ian MacDonald Apr 21 '15 at 21:59
  • @Ian MacDonald see myanswer it explains everything this puzzle isnt a duplicate orelse the bounty awarded answer should fit the bill – Abr001am Apr 21 '15 at 22:06
  • when i say unique, it doesnt mean unique for an empty grid, it means unique for any instantly filled data in blanks, that arnt contradictory for eachother. – Abr001am Apr 21 '15 at 22:33
  • Such a sudoku is not necessarily solvable if the only requirement you place on it be that the empty cells not hidden by the net can be filled in such that they themselves do not break the basic rules of sudoku. – Ian MacDonald Apr 22 '15 at 03:11
  • @IanMacDonald i knew , and wanted min blanks which they r unisolvable without any anomaly that would appear afterwards. v u felt the difference or not yet ? – Abr001am Apr 22 '15 at 10:05
  • in case of non-existence of such net , make a proof it doesnt , instead of anticipating any emo-based clueless proceedings , such proof would be benefic not just for me, but for all community – Abr001am Apr 22 '15 at 10:11
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    It's too bad the question got closed because now I finally understand what you mean I think. And the solution to that question would be interesting. I still think your wording is still not entirely clear because you need to define what "proper values" exactly means. It could mean a lot of things. You mean that the numbers should be unique in their 3x3 grid and columns and rows I believe. But "proper values" could also mean that the filled in numbers need to have a unique solution if you try to solve it with those numbers. – Ivo Apr 22 '15 at 12:01
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    @Agawa001: As worded, the question is a duplicate of the linked question. This is the reason it was closed. If you are certain that it is not a duplicate, you must work to clarify your question. As it stands now, it is unclear what you are asking (and could be closed yet again for this different reason). There are at least two valid acceptable reasons for which this question has been closed, neither of which are a personal attack on you. – Ian MacDonald Apr 22 '15 at 18:00
  • @Agawa001 I agree with you that this isn't a duplicate! Anyway, I also want to give you an advice for future posts: when you post a question/answer, try to be as clear as possible, especially when your post is likely to be misunderstood and considered a duplicate.
    Please, don't take my advice as an offence, I'm trying to give you a constructive opportunity to improve the global quality of your posts.     – leoll2 Apr 23 '15 at 14:06
  • @leoll2 my wording is fine , if i ever change the content of my puzzles it isnt because of its actual modality but it would rather be due to community requirements – Abr001am Apr 23 '15 at 17:52
  • @Agawa001: Does my edit capture your meaning? You were missing a significant number of definitions (and you attempted to redefine what "solvable" meant for a sudoku grid). – Ian MacDonald Apr 23 '15 at 18:21
  • @IanMacDonald i dont really get how did you adjust the globality of the problem in general , but , if the community approve that as equivalent sens of my previous version , take my agreement – Abr001am Apr 23 '15 at 18:30
  • @Agawa001: I have trouble understanding your most recent comment. If my edit to your puzzle does not match your original intention, revert the change. Only you know what you originally intended, so only you can be the judge of whether it matches your original intention. – Ian MacDonald Apr 23 '15 at 18:39
  • you may edit it simple way. it is complicated for any simple member to understand and deal with the actual form of puzzle , it remains my opinion – Abr001am Apr 23 '15 at 18:43
  • You want the number of blocked cells to be the minimum and allow some configurations not to be solvable? – leoll2 Apr 23 '15 at 19:25
  • not unsolvable for all arangements it cant be possible , i edited my problem lastly to require upper or lower bound of possible blocks which cant be fruitful all the time . – Abr001am Apr 23 '15 at 19:33
  • @leoll2 is it now clear what is demanded ? – Abr001am Apr 23 '15 at 19:47
  • Your wording is still unclear. Your definitions are inconsistent; for example, sudoku-friendliness is defined in a way that depends on what values the cells take, but sudoku nets are defined in terms of their visible sets being sudoku-friendly, despite not having any values. It's unclear why the solutions wouldn't simply be the all-allowed and all-blocked nets; the all-allowed net produces an invalid puzzle with a grid full of 1s, and the all-blocked net never produces a valid puzzle, since it has too many solutions. – user2357112 Apr 26 '15 at 01:58

2 Answers2

4

MAXIMUM BLOCKED BOXES:

$73$

A B X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X A B
----------------------------
X X X | A X X | X X X
X X X | B X X | X X X
X X X | X X X | X X X
----------------------------
X X X | X X A | X X X
X X X | X X B | X X X
X X X | X X X | X X X

Let's define cells C[row,column].
With this configuration, C[2,5] should contain both the values A and B, and this is impossible!

MINIMUM BLOCKED BOXES:

$1$

A B O | O O O | O O O
O O O | O X O | O O O
O O O | O O O | O A B
----------------------------
O O O | A O O | O O O
O O O | B O O | O O O
O O O | O O O | O O O
----------------------------
O O O | O O A | O O O
O O O | O O B | O O O
O O O | O O O | O O O

The X is the only blocked cell, O cells have generic values, while A and B are two different values. If you only block one cell, like in this case, there exists at least one configuration (as the one showed here) which makes the sudoku unsolvable.

BOUNTY QUESTION:

NO!

It's very easy to prove: take any grid $G$ and generate the sudoku-friendly subsets $S_1$ and $S_2$ blocking $80$ cells, leaving visible respectively C[1,1] and C[1,2].

This is $S_1$:

O X X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X X X
----------------------------
X X X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X X X
----------------------------
X X X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X X X

And this is $S_2$:

X O X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X X X
----------------------------
X X X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X X X
----------------------------
X X X | X X X | X X X
X X X | X X X | X X X
X X X | X X X | X X X

As you can see, both are legal patterns (ergo, sudoku-friendly), though they don't have a unique solution since a single information is never enough to solve univocally a sudoku!

Community
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leoll2
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  • so this would be the minimum blocks net right ? – Abr001am Apr 23 '15 at 20:01
  • yes , i think you misunderstood , both grids are solvable (not unique solutions but still solvable) see my example here of a non solvable grid with a friendly-sudoko inside – Abr001am Apr 23 '15 at 20:23
  • @Agawa001 Is the answer correct now? – leoll2 Apr 24 '15 at 16:51
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    well this is smart !!! and dont know yet if it its optimal , gonna let the community judge – Abr001am Apr 24 '15 at 17:01
  • @Agawa001 I've noticed that your account was suspended with a pending bounty. I'm curious to know what happens to the bounty in these situations, let me know if you find it out! – leoll2 May 02 '15 at 19:44
-1

for the minimum , i would choose this

7## | 123 | 456

#1# | 456 | 789

##4 | 789 | 123

------|------|------

147 | 938 | 562

258 | 671 | 394

369 | 245 | 817

------|------|------

471 | 562 | 938

582 | 394 | 671

693 | 817 | 245

Abr001am
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