# Sudoku-net pseudo-solvable with minimum blocks

• A sudoku net is defined here.

A sudoku net is defined as 'pseudo-solvable' if we can deduce at least two valid 'not unique' solutions.

• friendly sudoku is defined here

What is the sudoku net with the minimum possible blocks in such a way that this sudoku net is pseudo-solvable with at least one friendly sudoku?

Challenge 2

Find a sudoku net $$N$$ with visible set $$V_N$$ such that for any sudoku $$S_1$$, there is another $$S_2\ne S_1$$ where $$V_N(S_1) = V_N(S_2)$$ and $$|V_N|$$ is maximal.

Bonus question: (it is worth 50pts) is there any condition of pattern arrangements inside blanks which makes a sudoku unsolvable using previous result as work-domain?

• Can you clarify the order of quantifiers? (Something like: FIND a net such that FOR ALL solved sudokus, if you place the net over the sudoku, then THERE EXISTS at least two solutions to the resulting puzzle) Commented Apr 24, 2015 at 16:24
• @Lopsy in fact its not for all arrangements , there is specific oredering which would work it out Commented Apr 24, 2015 at 16:29
• @Lopsy in that case such sudoko can be with 80 blocks so trivial for a question dont you think ? Commented Apr 24, 2015 at 16:40
• @Lopsy does it look pleasant or encouraging now ? Commented Apr 24, 2015 at 16:57
• Honestly I'm still not 100% sure what you mean, and I wouldn't want to work on a question like this one only to be told that I solved the wrong puzzle. Is your first question equivalent to "What's the smallest possible number of cells you can change to get from one solved Sudoku to a different solved Sudoku?" Commented Apr 24, 2015 at 17:11

# MINIMUM BLOCKED BOXES:

$$4$$

O O A | O O O | C O O
O O O | O O O | O O O
O O B | O O O | D O O
-------------------------------
O O O | O O O | O O O
O O O | O O O | O O O
O O O | O O O | O O O
-------------------------------
O O O | O O O | O O O
O O O | O O O | O O O
O O O | O O O | O O O

Where A,B,C,D are equal in pairs (the O are given numbers).
Suppose that both the top-left and the top-right squares are missing the numbers 5 and 7. If you swap A with B, and C with D the sudoku will be still valid. This means that if you have clues about all cells except A,B,C,D there are 2 different valid solutions to the sudoku, which means UNSOLVABLE.

Given a random solved sudoku, what's the minimal net that makes it pseudo-solvable?
The answer is: it depends on the sudoku! Some particular sudoku need a net with just $$4$$ blocks, like the one showed in the picture, while other require up to $$18$$ blocks to become pseudo-solvable (put the blocks on all $$1$$ and $$2$$). Example of the latter configuration is here:

If you want a net that works for any sudoku, independently from the grid, just put the blocks on the first and second row. Anyway, this would still require $$18$$ blocks, not better than my previous solution.

# When is a sudoku pseudo-solvable?

When you don't have information about $$4$$ cells, arranged on a rectangular shape, such that $$2$$ of them belong to a quadrant, while the other $$2$$ belong to another quadrant (as shown in the above picture); also, the cells on the opposite vertices of the rectangle should contain the same values!

Let's say these values are $$a$$ and $$b$$ and the pattern is the same described in the above picture. The first row says the A and C must contain $$a$$ and $$b$$, but doesn't say in which order. Same for the third row, third column, seventh column, top-left box, top-right box. This means, there are $$2$$ valid solutions for this sudoku.

# When is a sudoku NOT pseudo-solvable?

This means solvable or impossible! A Sudoku is impossible when it has internal contradictions (like two $$7$$ in the same row). When the sudoku isn't impossible (aka possible), it can be either solvable (admits 1 solution) or pseudo-solvable (admits 2+ solutions). How to tell them? First of all, given a grid, fill as many cells as possible using deductions. Now, if you have 4 unknown cells arranged in a rectangular pattern (see above for the other conditions), then the sudoku is pseudo-solvable!

• this is correct for the first question . Commented Apr 24, 2015 at 16:55
• @Agawa001 Added answer for the second, thinking about the bonus. Commented Apr 24, 2015 at 17:02
• i changed the content btw , but your first intervention is still valid Commented Apr 24, 2015 at 17:03
• leoll2 refering to your latest edit , do you mean the shortest chain ? Commented Apr 24, 2015 at 18:11
• @Agawa001 my last edit is the answer to the 2nd question, I provided the minimum number of blocks in a net that transforms a solved sudoku into a pseudo-solvable one! Commented Apr 24, 2015 at 18:32

my answer for the second question may be :

# # # | # # # | # # #

# # # | # # # | # # #

____ | ____ | ____

____ | ____ | ____

____ | ____ | ____

____ | ____ | ____

____ | ____ | ____

____ | ____ | ____

____ | ____ | ____

18 blocks

bonus question :

any logical relation isnt known yet but sure ... it would have something to do with graph-theory

• That's 18 blocks, not 12.
– user88
Commented Apr 24, 2015 at 19:31
• fixed it ................ Commented Apr 24, 2015 at 19:35
• So, 18 blocks is the optimal net, as in my post? Commented Apr 24, 2015 at 19:37
• dunno yet :S ... im waiting more initiatives Commented Apr 24, 2015 at 19:38