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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 ?
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22
  • $\begingroup$ Please describe how this maximal net differs from the maximal net in the linked question. $\endgroup$
    – Ian MacDonald
    Commented Apr 21, 2015 at 20:32
  • $\begingroup$ i can still roll it back to first non edited version if it is duplicate $\endgroup$
    – Abr001am
    Commented Apr 21, 2015 at 20:36
  • $\begingroup$ @Ian MacDonald the solution shown in previous puzzle doesnt fit my puzzle $\endgroup$
    – Abr001am
    Commented Apr 21, 2015 at 20:37
  • 1
    $\begingroup$ 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. $\endgroup$
    – Ivo
    Commented Apr 22, 2015 at 12:01
  • 1
    $\begingroup$ @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. $\endgroup$
    – Ian MacDonald
    Commented Apr 22, 2015 at 18:00

2 Answers 2

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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!

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5
  • $\begingroup$ so this would be the minimum blocks net right ? $\endgroup$
    – Abr001am
    Commented Apr 23, 2015 at 20:01
  • $\begingroup$ 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 $\endgroup$
    – Abr001am
    Commented Apr 23, 2015 at 20:23
  • $\begingroup$ @Agawa001 Is the answer correct now? $\endgroup$
    – leoll2
    Commented Apr 24, 2015 at 16:51
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    $\begingroup$ well this is smart !!! and dont know yet if it its optimal , gonna let the community judge $\endgroup$
    – Abr001am
    Commented Apr 24, 2015 at 17:01
  • $\begingroup$ @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! $\endgroup$
    – leoll2
    Commented May 2, 2015 at 19:44
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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

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