The New Sudoku Players' Forum

[coloin]

revisiting the 2 clue in a band scenario .... which has not been totally clear

Code: Select all

        p123
+-----+-----+-----+
|. . .|. 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|
+-----+-----+-----+

tells us that
029
199
199 is invalid

therefore
033
199
199 is needed

Code: Select all

       p110
+-----+-----+-----+
|. . .|. . 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|
+-----+-----+-----+

099
119
199 is invalid

therefore
033
122
122 is at least required - as concluded by serg above ......

however

Code: Select all

        P114             
 +-----+-----+-----+     
 |. . .|. 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|     
 +-----+-----+-----+

means that

033
132
123

or

033
132
132

is a requirement

and here is one example of a puzzle of the first pattern

Code: Select all

+---+---+---+
|...|...|2..|
|...|498|...|
|...|...|.15|
+---+---+---+
|...|9..|...|
|...|.52|...|
|.8.|...|.36|
+---+---+---+
|...|.8.|59.|
|...|...|...|
|2..|..6|..7|
+---+---+---+  18 clues

and here is one example of a puzzle of the second pattern

Code: Select all

+---+---+---+
|...|...|...|
|...|345|...|
|...|...|671|
+---+---+---+
|...|9..|...|
|...|.16|...|
|.2.|...|.84|
+---+---+---+
|...|4..|.9.|
|...|.27|...|
|1..|...|..5|
+---+---+---+  18 clues

I have now more than 50 18-puzzles with these 2 patterns .....

We can say now that there cannot be a 17-puzzle with 2 clues in a band.

c

[Serg]

Hi, coloin!

revisiting the 2 clue in a band scenario .... which has not been totally clear

Code: Select all

        p123
+-----+-----+-----+
|. . .|. 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|
+-----+-----+-----+

tells us that
029
199
199 is invalid

Right.

therefore
033
199
199 is needed

Why?

therefore

Code: Select all

       p110
+-----+-----+-----+
|. . .|. . 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|
+-----+-----+-----+

099
119
199 is invalid

Right.

therefore
033
122
122 is at least required - as concluded by serg above ......

Why? When did I say it?

however

Code: Select all

        P114             
 +-----+-----+-----+     
 |. . .|. 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|     
 +-----+-----+-----+

means that

033
132
123

or

033
132
132

is a requirement

I don't understand your point. Please, clarify your considerations.

Serg

[coloin]

I agree with your logic. If a map has band 011 (i.e. 1 empty box and 2 boxes containing exactly 1 clue each), then each remaining band must have 322 configuration, at least....

033
122
122 has 16 clues

I have only extended it to show that

033
123
132 has 18 clues and has valid puzzles

033
122
132 has 17 clues and has no valid puzzles

this shows that in patterns with 2 clues only in a band
- there are no 17-puzzles
- there are no 9plus14 puzzles [9 clues in central box]

it doesnt change the remaining maps needed to search for {3+4} clues in two bands [as already {2+5} is disproved]

contemplating an exaustive grid solution search might be reduced by
only search grids which have a band completable in 3 clues - and we know the clues

c

[dobrichev]

contemplating an exaustive grid solution search might be reduced by
only search grids which have a band completable in 3 clues - and we know the clues

But a band completable in 2 clues is easy to complete with 3 clues, right?

[Serg]

Hi, coloin!

I agree with your logic. If a map has band 011 (i.e. 1 empty box and 2 boxes containing exactly 1 clue each), then each remaining band must have 322 configuration, at least....

033
122
122 has 16 clues

Yes, it was my point. Sorry, I fogot my own considerations .

I have only extended it to show that

033
123
132 has 18 clues and has valid puzzles

033
122
132 has 17 clues and has no valid puzzles

I agree with you. Your propositions are right.

this shows that in patterns with 2 clues only in a band
- there are no 17-puzzles

Yes! Impressive expectation! But you should add (for the sake of completeness) evident consideration - band having 2 empty boxes and the third box containing 2 clues cannot participate valid puzzles.

- there are no 9plus14 puzzles [9 clues in central box]

Sorry, I don't catch your idea, please post your conclusion path in details.

it doesnt change the remaining maps needed to search for {3+4} clues in two bands [as already {2+5} is disproved]

Surely.

contemplating an exaustive grid solution search might be reduced by
only search grids which have a band completable in 3 clues - and we know the clues

Yes, it is right thought!(It is assumed that band configurations 012 and 111 should be checked while searching for new 17-clue puzzles, but band configuration 003 (having 3 clues though) is impossible in any valid puzzles.)

Good work!

Serg

[coloin]

But a band completable in 2 clues is easy to complete with 3 clues, right?

welcome to the discusssion, indeed the sole band which solves with 2 clues has many combinations which solve with 3 clues.
however unless we get rid of many grids its all hypothetical
its quite possible that most grids have a band solvable with 3 clues - given that 100/416 bands are solvable in 3 - from your work here

splitting it up into just the ED 2 band [54 clue] combinations might be a potential way
but not sure how many million there are [? 500 million !]
all those not having a band completable with 3 clues can be discarded ..... this gets rid of ~ 42% [assuming band freq is similar.]
the other band will need to be solvable in 4 clues - this excludes another ~8% [58/416 bands need more than 4 clues]

Yes! Impressive expectation! But you should add (for the sake of completeness) evident consideration - band having 2 empty boxes and the third box containing 2 clues cannot participate valid puzzles.

yes
099
099
299 is invalid

you showed that this pattern did not have valid puzzles

Code: Select all

+-----+-----+-----+
|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|
+-----+-----+-----+

Sorry, I don't catch your idea, please post your conclusion path in details.

well i have been trying to generate 9plus12-puzzles
it probably precipitated this thread ....
at least 4 clues in any one band is the norm. i probably have found all but a few 9plus12s, and many of the 9plus13s.
C

[coloin]

Reopening this thread as I have investigated a little more

Code: Select all

Adding 6 clues is never enough ?
 
9+---+---+---+
 |...|...|..x|
 |...|...|..x|
 |...|...|..x|
 +---+---+---+
 |...|...|..x|
 |...|...|..x|
 |...|...|..x|
 +---+---+---+
 |...|...|xxx|
 |...|...|xxx|
 |xxx|xxx|xxx|
 +---+---+---+

3 examples of a valid puzzle with 7 more clues

Code: Select all

                                                                   
+---+---+---+         +---+---+---+         +---+---+---+           
|123|456|789|         |123|456|789|         |123|456|789|           
|456|...|...|         |456|...|...|         |457|...|.2.|           
|789|...|...|         |789|...|...|         |869|...|...|           
+---+---+---+         +---+---+---+         +---+---+---+           
|2..|9..|.3.|         |2..|.18|...|         |2..|8..|.5.|           
|5..|3..|...|         |6..|...|5..|         |3..|...|...|           
|8..|...|6..|         |8..|...|..4|         |6..|.9.|...|           
+---+---+---+         +---+---+---+         +---+---+---+           
|3..|..7|...|         |3..|..9|.2.|         |5..|...|..6|           
|6..|.2.|5..|         |5..|...|...|         |7..|..3|.4.|           
|9..|...|...|         |9.1|...|...|         |9..|...|...|           
+---+---+---+ 816     +---+---+---+  202    +---+---+---+  3       

I searched and found only 1021 essentially different of these 21plus7 puzzles
None could be reduced to a 21plus6 - not proof but very unlikely to be found.

816/1021 puzzles had the repeating minirow band in both directions.
202/1021 puzzles had the repeating minirow band in one direction.
3/1021 puzzles had no repeating minirow band in either direction.

here is 200 of them

Hidden Text: Show

Code: Select all

123456789456......789......2........3......258..7.9...5...4..1.6.7......9........
123456789456......789......2........5.......48..9..6..3...25...6....1...9......5.
123456789456......789......2........5.......68..2.7...3..1..2..6....9...9.....3..
123456789456......789......2........5......218..9...3.3...1....6.....5..9..7.....
123456789456......789......2........5......238...7....3....2..56.......49...8....
123456789456......789......2........5......4.8..9....63...25.6.6...1....9........
123456789456......789......2........5......638..1.....3..2..1.76........9.......5
123456789456......789......2........5......688..1....43..2..1..6........9.......5
123456789456......789......2........5.....3..8..7....13......2.6.....5..9..8.3...
123456789456......789......2........5.....3..8..7....43......2.6.....5..9..8.3...
123456789456......789......2........5.....41.8..9.2...3........6.....52.9...1....
123456789456......789......2........5.....42.8..39....3........6...4.1..9..2.....
123456789456......789......2........5.....6.38..7.....3........6...23..19.....4..
123456789456......789......2........5.....6.38..74....3......7.6....1...9.....4..
123456789456......789......2........5.....6.38..74....3....1...6......7.9.....4..
123456789456......789......2........5.....9..8...21...3...4..2.6.......19..7.....
123456789456......789......2........5.....9..8..3....43...7..2.6.......59..1.....
123456789456......789......2........5....1..38..3...6.3..9.....6...8.2..9........
123456789456......789......2........5....1..38..3..6..3..9.....6...8..2.9........
123456789456......789......2........5....3.1.8.......43..8..2..6...4....9...7....
123456789456......789......2........5....36..8......7.3.......86..7.....9...8..2.
123456789456......789......2........5....36..8....4...3.......86..7.....9...8..2.
123456789456......789......2........5....4...8..2...133........6.....4..9...8.6..
123456789456......789......2........5....7..48...9....3....1...6..3.....9......25
123456789456......789......2........5....96..8......9.3.....4.16...2....9...8....
123456789456......789......2........5....96..8......9.3..2..4..6.......19..8.....
123456789456......789......2........5....96..8..7.....3.....4.16....8...9...2....
123456789456......789......2........5....98..8..7.....3.....4.16....8...9...2....
123456789456......789......2........5...1...48.....6..3.....1..6.....4..9...3..2.
123456789456......789......2........5...1...48..32....3......2.6....8..59........
123456789456......789......2........5...1..4.8.....6..3.....1..6....2...9....8..2
123456789456......789......2........5...1.6.48........3..8...2.6....7.9.9........
123456789456......789......2........5...1.6.48.....9..3..8...2.6....7...9........
123456789456......789......2........5...1.9..8...2.6..3......2.6....8...9..7.....
123456789456......789......2........5...1.9.48.....6..3..8...2.6....7...9........
123456789456......789......2........5...14...8......6.3..9....56.....4..9...8....
123456789456......789......2........5...17...8.......33...29...6.....1..9.....4..
123456789456......789......2........5...17...8.......33..2.9...6.....4..9.....1..
123456789456......789......2........5...2...48.......73.....2..6..9.8...9..7.....
123456789456......789......2........5...27...8.....6.33...89...6.....4..9........
123456789456......789......2........5...27...8.....6.33...98...6.....4..9........
123456789456......789......2........5...27...8.....63.3....8...6..1..4..9........
123456789456......789......2........5...3....8....14..3.......56......329....4...
123456789456......789......2........5...3....8....16..3.......26....4...9......35
123456789456......789......2........5...3....8....74..3.......56......329....4...
123456789456......789......2........5...3....8..9....43.....2..6.....5..9..8.3...
123456789456......789......2........5...3....8..9....73.....2..6.....53.9..8.....
123456789456......789......2........5...3....9..8....73.....2..6.....53.8..7.....
123456789456......789......2........5...3...68...1....3.....2..6....8...9..7....3
123456789456......789......2........5...4.6..8..7.....3.....4.16........9...82...
123456789456......789......2........5...6.8..8..7.....3.....4.16...8....9....2...
123456789456......789......2........5...7....8......363....8...6..2.....9.....5.7
123456789456......789......2........5...7....8......463...1....6........9..83.5..
123456789456......789......2........5...7....8......643...1....6........9..83.2..
123456789456......789......2........5...7....8.....6233.....4..6...8....9....2...
123456789456......789......2........5...7....8.....6323.....4..6...8....9....2...
123456789456......789......2........5...7....8....9.3.3....8...6.....5..9..23....
123456789456......789......2........5...7....8...9..3.3....2...6.......49.....5.7
123456789456......789......2........5...7....8..1...323...8....6.....5..9..2.....
123456789456......789......2........5...7....8..9.1.3.3....2...6.....5..9.......4
123456789456......789......2........5...7.1..8..9...6.3...8....6....24..9........
123456789456......789......2........5...7.6..8......3.3....2...6.....4..9..38....
123456789456......789......2........5...71...8...9...23.....1..6.....4..9...2....
123456789456......789......2........5...74...8......3.3...8....6..3..4..9.....1..
123456789456......789......2........5...9...78....1...3...82...6.....1.49........
123456789456......789......2........5..1.....8.......63..2..1..6....9...9...8..2.
123456789456......789......2........5..1....68..3.....3.....8..6....9...9...8..2.
123456789456......789......2........5..1...4.8.....6..3.....1..6....2...9....8..2
123456789456......789......2........5..1..3.48...9...63..7..1..6........9........
123456789456......789......2........5..1..4..8......363.....17.6........9....8...
123456789456......789......2........5..1..4..8....3..63.....12.6...4....9........
123456789456......789......2........5..1..4..8....9...3..7..15.6........9...8....
123456789456......789......2........5..1..4..8...9...63..7..1..6........9...8....
123456789456......789......2........5..1..4..8..3....63.....2..6....9...9...8....
123456789456......789......2........5..1..4..8..3....63.....2..6...9....9...7....
123456789456......789......2........5..14....8...3...63.....4..6....9...9......3.
123456789456......789......2........5..17....8......363..2...1.6........9...8....
123456789456......789......2........5..17....8.....3.63..2...1.6........9...8....
123456789456......789......2........5..17....8...3...63.....1..6....9...9......3.
123456789456......789......2........5..2..3..8..3....63...7.1..6...9....9........
123456789456......789......2........5..24....8.....6.33........6...7.4..9....8...
123456789456......789......2........5..27....8.....6.33.....4..6...8....9....2...
123456789456......789......2........5..27....8.....63.3.....4..6...8....9....2...
123456789456......789......2........5..3..4..8...9...63..2..1..6........9...8....
123456789456......789......2........5..31.4..8.....6..3......1.6.....3..9....8...
123456789456......789......2........5..7.....8...9..3.3....2...6.....41.9.......7
123456789456......789......2........5..7.....8..9.1.3.3........6....25..9.......4
123456789456......789......2........5..7....48....1...3......2.6.....5..9..8.3...
123456789456......789......2........5..7....68..3.....3.....8..6....9...9...8..2.
123456789456......789......2........5..7..1..8......363.....41.6........9....8...
123456789456......789......2........5..7..1..8...9...63..1..2..6........9...8....
123456789456......789......2........5..7..2..8.......33...98...6.....5..9......4.
123456789456......789......2........5..7..2..8......563.....4..6...98...9........
123456789456......789......2........5..7..4..8...9...63..1..2..6........9...8....
123456789456......789......2........5..71....8......363...4..1.6........9..8.....
123456789456......789......2........5..71....8.....3.63...4..1.6........9..8.....
123456789456......789......2........5..72....8.....63.3....8...6..1..4..9........
123456789456......789......2........5..74....8......363.....5..6..9.....9....2...
123456789456......789......2........5..74....8......363...2..1.6........9..8.....
123456789456......789......2........56...7...8......323.....1..6....8...9...2....
123456789456......789......2........56..7....8......323.....1..6...8....9..2.....
123456789456......789......2........6.....5.28....9...3...2....5..16.4..9........
123456789456......789......2.......15........8..9...6.3.....2..6.....5..9..8.3...
123456789456......789......2.......15..7.....8..9....33....25..6......4.9........
123456789456......789......2.......35........8....74..3..8.1...6....2...9.......5
123456789456......789......2.......35........8...4.6..3.....4..6.....1..9..2.3...
123456789456......789......2.......35....4...8....1...3...7....6.....1..9...8..2.
123456789456......789......2.......35....7...8....4.9.3.....5..6......7.9..2.....
123456789456......789......2.......35....7...8....9.6.3.....5..6......7.9...2....
123456789456......789......2.......35....74..8......9.3........6.....1..9..23....
123456789456......789......2.......35...4....8.....62.3........6...7.4..9....2...
123456789456......789......2.......35...7....8.....62.3........6...1.4..9....2...
123456789456......789......2.......35...71...8......2.3..89....6.....4..9........
123456789456......789......2.......35...74...8.....6..3.....4..6....2...9...8....
123456789456......789......2.......35...9....8.....6.23.....4..6...82...9........
123456789456......789......2.......35..1.4...8......9.3.....4..6.......19...8....
123456789456......789......2.......35..17....8......2.3..98....6.....4..9........
123456789456......789......2.......35..17....8......6.3..98....6.....4..9........
123456789456......789......2.......35..29....8.....6..3.....4..6...82...9........
123456789456......789......2.......35..29....8.....6..3..1..4..6...2....9........
123456789456......789......2.......35..29....8.....6..3..1..4..6...7....9........
123456789456......789......2.......35..29....8.....6..3..1..4..6...8....9........
123456789456......789......2.......35..29....8.....6..3..1..5..6...8....9........
123456789456......789......2.......35..7.....8......2.3......7.6....25..9....3...
123456789456......789......2.......35..7.....8.....69.3.....4..6........9...23...
123456789456......789......2.......35..7..6..8......9.3.....4..6........9...32...
123456789456......789......2.......35..92....8.....6..3...1.4..6..5.....9........
123456789456......789......2.......35..92....8.....6..3...1.5..6....8...9........
123456789456......789......2.......356...4...8........3.....4..6.....5..9..32....
123456789456......789......2.......356.7.....8........3.....5..6.....4..9...32...
123456789456......789......2.......45........6....7...3..2..5..8......7.9..14....
123456789456......789......2.......45..6..1..8...9....3...8....6.....4..9....2...
123456789456......789......2.......53..8.....8....7...5......146....8...9...2....
123456789456......789......2.......55.....6..8..7.....3.....4.16....8...9...2....
123456789456......789......2.......55.....6..8..7.....3.....4.16....9...9...2....
123456789456......789......2.......55.....6..8..7.....3..1...4.6...2.8..9........
123456789456......789......2.......55...3....8......6.3.....2..6....9...9....41..
123456789456......789......2.......55..1..4..8....3...3....71..6........9...8....
123456789456......789......2.......65........8....74..3....2...6..8.1...9.......5
123456789456......789......2.......65........8....74..3..8.1...6....2...9.......5
123456789456......789......2.......65.....2..8..9.....3...721..6....1...9........
123456789456......789......2.......65.....23.8...7....3.......46...1....9....2...
123456789456......789......2.......65.....23.8..7.....3...8....6.......49....2...
123456789456......789......2.......65.....3..8...17...3...7..1.6....9...9........
123456789456......789......2.......65...1.2..8..7.....3.......46......7.9....2...
123456789456......789......2.......65...47...8....9.3.3.....1..6........9...2....
123456789456......789......2.......65...7....8........3...8.1..6.......49..2.3...
123456789456......789......2.......65...7....8......233.....4..6...1....9....2...
123456789456......789......2.......65...7....8......3.3....2...6.......49..38....
123456789456......789......2.......65...7....8......3.3....2...6.....3.49...8....
123456789456......789......2.......65...7....8......323.....6..6...9....9....2...
123456789456......789......2.......65...7....8..9.....3....21..6...8....9......4.
123456789456......789......2.......65...7....8..9.....3....25..6.....2..9..6.....
123456789456......789......2.......65...74...8....1...3........6.....1..9...8..3.
123456789456......789......2.......65..14....8...3....3.....1..6....9...9......3.
123456789456......789......2.......65..14....8..3.....3.....1..6....9...9......3.
123456789456......789......2.......65..17....8.......33.......16.....4..9...8....
123456789456......789......2.......65..7..3..8...9....3..1..2..6........9...8....
123456789456......789......2.......65..7..4..8...9....3..1..2..6........9...8....
123456789456......789......2.......65..71....8.......33.......16.....5..9..8.....
123456789456......789......2.......65..8.....8...7....3......646....2...9...8....
123456789456......789......2.......65..8.....8...9....3....1..46....2...9.....1..
123456789456......789......2.......65..8.....8..7...2.3....2...6...8....9.......4
123456789456......789......2.......65..8.....8..7..2..3....2...6...8....9.......4
123456789456......789......2.......65..8.....8..7..3..3....2...6...8....9.......4
123456789456......789......2.......65..8.....8..7..4..3....2...6...8....9.......4
123456789456......789......2.......65..8..1..8..7.....3....2...6...8....9.......4
123456789456......789......2.......65..89....8......3.3...8....6....2...9.......4
123456789456......789......2.......65..89....8.....3..3...82...6........9.......5
123456789456......789......2.......75...7.1..8..9.....3....24..6...8....9........
123456789456......789......2.......85..7.....8......563..1..4..6....8...9........
123456789456......789......2......1.5.......48...9....3..7.....6..1.....9......25
123456789456......789......2......1.5.....6..8...7....3.....4.66...2....9...8....
123456789456......789......2......1.5....3..68....2...3..8.....6......2.9...7....
123456789456......789......2......1.5....36..8....2...3..8.....6......2.9...7....
123456789456......789......2......1.5...8....8...9..3.3....2...6.......49.....5..
123456789456......789......2......1.5..7.....8..9...3.3....2...6.......49.....5..
123456789456......789......2......135...7....8........3.....4..6....25..9....3...
123456789456......789......2......135..7.....8........3.....4..6....25..9....3...
123456789456......789......2......143........8..7.9...5...4..3.6.7......9........
123456789456......789......2......143..7.9...8........5...1..3.6.7......9........
123456789456......789......2......143..7.9...8........5...4..3.6.7......9........
123456789456......789......2......153........8..7.9...5...4..3.6.7......9........
123456789456......789......2......153...79...8........5..1...3.6.7......9........
123456789456......789......2......153...79...8........5..1...4.6.7......9........
123456789456......789......2......153..7.9...8........5...4..3.6.7......9........
123456789456......789......2......3.5.....6..8...7....3.....4.66...2....9...8....
123456789456......789......2......3.5....4...8...71...3.....4..6........9...8...5
123456789456......789......2......3.5...1...48...2....3......2.6..9.....9....8...
123456789456......789......2......3.5...7....8.....6.23.....4..6...8....9....2...
123456789456......789......2......3.5...7....8....3.6.3........6..2.....9.....4.5
123456789456......789......2......3.5...71...8.......63.....4..6........9...8..7.
123456789456......789......2......3.5...74...8........3...8....6..2..4..9.....1..
123456789456......789......2......3.5...74...8.....6..3.....4..6....2...9...8....
123456789456......789......2......3.5..17....8.......63....74..6...8....9........
123456789456......789......2......315..7.....8........3....25..6......2.9....1...
123456789456......789......2......315..7.....8........3....25..6....1...9.....4..
123456789456......789......2......345....1...8...7....3.....4..6........9...8...5
123456789456......789......2......353........8..7.9...5...4..1.6.7......9........
123456789456......789......2......365........8...47...3...7..1.6....8...9........

[JPF]

And these are ? unproven patterns with n clues to be added - in order of possible ease !
Probably they are all correct ! Apologies if they have indeed been proven in former posts.....

The fact that we havnt got proof probably means that it is not an easy task !

Code: Select all

Adding 4 clues is never enough ?                                     
                                                                                                                                     
 7+---+---+---+       +---+---+---+                                 
  |...|...|...|       |...|...|781|                                 
  |...|...|...|       |96.|...|...|                                 
  |...|...|...|       |...|...|...|                                 
  +---+---+---+       +---+---+---+                                 
  |...|xxx|...|       |...|123|...|                                 
  |...|xxx|...|       |...|456|...|                                 
  |...|xxx|...|       |...|789|...|                                 
  +---+---+---+       +---+---+---+                                 
  |xxx|...|xxx|       |659|...|317|                                 
  |xxx|...|xxx|       |437|...|268|                                 
  |xxx|...|xxx|       |821|...|459|                                 
  +---+---+---+       +---+---+---+    valid plus 5 clues                                                                                                                                                                                                                                                     

actually, adding 4 clues may be enough:

Code: Select all

+---+---+---+
|.3.|..2|...|
|...|...|..4|
|...|...|..5|
+---+---+---+
|...|123|...|
|...|456|...|
|...|789|...|
+---+---+---+
|195|...|348|
|284|...|756|
|376|...|219|
+---+---+---+


JPF

[JPF]

Code: Select all

Adding 4 clues is never enough ?                                     
                                                                                           
 8+---+---+---+       +---+---+---+                                 
  |...|...|..,|       |.2.|...|...|                                 
  |...|...|..,|       |...|.5.|9..|                                 
  |...|...|..,|       |...|...|...|                                 
  +---+---+---+       +---+---+---+                                 
  |...|...|xxx|       |...|6..|123|                                 
  |...|...|xxx|       |..9|...|456|                                 
  |...|...|xxx|       |...|...|789|                                 
  +---+---+---+       +---+---+---+                                 
  |xxx|xxx|xxx|       |281|345|697|                                 
  |xxx|xxx|xxx|       |476|891|532|                                 
  |xxx|xxx|xxx|       |935|267|841|                                 
  +---+---+---+       +---+---+---+   valid  plus 5 clues 


here is one puzzle valid plus 4 clues:

Code: Select all

+---+---+---+
|...|...|...|
|...|1..|...|
|23.|...|...|
+---+---+---+
|...|.4.|562|
|...|...|731|
|...|...|849|
+---+---+---+
|678|921|453|
|451|837|926|
|923|465|178|
+---+---+---+


JPF

[coloin]

Excellent ..... those two puzzles are valid and show that there is [as ever] always incidences in the sudoku puzzle space which surprise us ! I will update the posts...
Evidently in these cases I did not search as hard as I should have ! Well done !
I wonder were there many examples that you found ? if there were very few found this would increase the likelihood that the limit was reached.

Of course as touched on before, there are the patterns plus n ( where n>1) clues as we have here , which are more of a generalization, rather than some of the specific patterns which are "maximal and have no puzzles" - whereby the addition of a single clue anywhere in the remaining cells now would give a valid puzzle.

I would be interested if you looked at the other cases which had very few puzzle examplars and whether you found any more !!!
eg

Code: Select all

+---+---+---+                                     
|123|456|789|                                     
|...|...|..1|                                     
|...|.9.|...|                                     
+---+---+---+                                     
|891|243|567|                                     
|4..|...|...|                                     
|...|...|9..|                                     
+---+---+---+                                     
|...|..1|...|                                     
|...|...|...|                                     
|965|872|413|                                     
+---+---+---+ valid plus 5 clues  [only one found]


[JPF]

Code: Select all

+---+---+---+                                     
|123|456|789|                                     
|...|...|..1|                                     
|...|.9.|...|                                     
+---+---+---+                                     
|891|243|567|                                     
|4..|...|...|                                     
|...|...|9..|                                     
+---+---+---+                                     
|...|..1|...|                                     
|...|...|...|                                     
|965|872|413|                                     
+---+---+---+ valid plus 5 clues  [only one found]


I'm not sure I understand your point.
I found easily more than 1000 valid puzzles with the same pattern.
Here are some examples:

Code: Select all

+---+---+---+   +---+---+---+   +---+---+---+
|123|456|789|   |123|456|789|   |123|456|789|
|...|...|..5|   |...|...|..1|   |...|...|..1|
|...|.9.|...|   |...|.8.|...|   |...|.8.|...|
+---+---+---+   +---+---+---+   +---+---+---+
|895|243|167|   |891|263|547|   |981|563|247|
|6..|...|...|   |4..|...|...|   |4..|...|...|
|...|...|2..|   |...|...|2..|   |...|...|5..|
+---+---+---+   +---+---+---+   +---+---+---+
|...|..1|...|   |...|..7|...|   |...|..1|...|
|...|...|...|   |...|...|...|   |...|...|...|
|914|872|653|   |768|592|413|   |865|792|413|
+---+---+---+   +---+---+---+   +---+---+---+


Here are 2 valid puzzles with different patterns:

Code: Select all

+---+---+---+   +---+---+---+
|123|456|789|   |123|456|789|
|...|...|...|   |...|...|...|
|5..|..2|...|   |5..|1..|...|
+---+---+---+   +---+---+---+
|341|827|695|   |839|625|147|
|...|...|...|   |...|...|...|
|7..|..9|...|   |.6.|.9.|...|
+---+---+---+   +---+---+---+
|692|145|873|   |458|739|216|
|...|...|...|   |...|...|...|
|...|...|1..|   |...|...|5..|
+---+---+---+   +---+---+---+


I'm pretty sure that this pattern is invalid:

Code: Select all

+---+---+---+   +---+---+---+
|xxx|xxx|xxx|   |123|456|789|
|...|...|...|   |...|...|...|
|x..|x..|x..|   |5..|1..|6..|
+---+---+---+   +---+---+---+
|xxx|xxx|xxx|   |879|621|543|
|...|...|...|   |...|...|...|
|.x.|...|...|   |.6.|...|...|
+---+---+---+   +---+---+---+
|xxx|xxx|xxx|   |417|893|256|
|...|...|...|   |...|...|...|
|...|...|x..|   |...|...|9..|
+---+---+---+   +---+---+---+
Min=2 sol.           2 sol.


JPF

[coloin]

I'm not sure I understand your point.....

Well your methods now are oviously superior to mine. The more examples of a pattern that you can make the more likely that there will be a superflous clue in the added clues [stating the obvious I suppose]
It was some time ago and perhaps methods have improved !!
We are well short of any proof however and the number and ease of puzzles found is probably a good guide to the lower limit
Have updated with your new finds ...

[JPF]

B1B5B9 + 3 clues
The only valid 3 empty box patterns are:

Code: Select all

00X   0XX
XX0   X0X
XXX   XX0


and here is the only valid 4 empty box pattern:

Code: Select all

X0X
0X0
X0X


Therefore for B1B5B9 + 3 clues, there exists only 5 possible ed-patterns:

Code: Select all

+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|XXX|...|...|   |XXX|...|...|   |XXX|...|...|   |XXX|...|...|   |XXX|...|...|
|XXX|...|...|   |XXX|...|...|   |XXX|.X.|...|   |XXX|...|..X|   |XXX|...|XX.|
|XXX|...|...|   |XXX|...|...|   |XXX|...|...|   |XXX|...|.X.|   |XXX|...|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|XXX|...|   |...|XXX|...|   |...|XXX|...|   |...|XXX|...|   |...|XXX|...|
|...|XXX|.X.|   |...|XXX|.X.|   |...|XXX|.X.|   |...|XXX|...|   |...|XXX|...|
|...|XXX|...|   |...|XXX|...|   |...|XXX|...|   |...|XXX|...|   |...|XXX|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|...|XXX|   |...|...|XXX|   |...|...|XXX|   |...|...|XXX|   |...|...|XXX|
|...|.X.|XXX|   |.X.|.X.|XXX|   |.X.|...|XXX|   |...|...|XXX|   |.X.|...|XXX|
|.X.|...|XXX|   |...|...|XXX|   |...|...|XXX|   |X..|...|XXX|   |...|...|XXX|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+


The first four patterns are valid.
Here is an example for each of them :

Code: Select all

+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|123|...|...|   |123|...|...|   |123|...|...|   |123|...|...|
|456|...|...|   |456|...|...|   |456|.7.|...|   |456|...|..7|
|789|...|...|   |789|...|...|   |789|...|...|   |789|...|.4.|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|347|...|   |...|769|...|   |...|213|...|   |...|324|...|
|...|986|.5.|   |...|453|.2.|   |...|967|.4.|   |...|157|...|
|...|512|...|   |...|218|...|   |...|845|...|   |...|689|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|...|897|   |...|...|247|   |...|...|861|   |...|...|325|
|...|.7.|341|   |.1.|.9.|856|   |.3.|...|259|   |...|...|819|
|.9.|...|562|   |...|...|319|   |...|...|473|   |3..|...|476|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+


The last pattern is probably invalid.
Here is a puzzle with 2 solutions:

Code: Select all

+---+---+---+
|123|...|...|
|456|...|23.|
|789|...|...|
+---+---+---+
|...|124|...|
|...|765|...|
|...|893|...|
+---+---+---+
|...|...|198|
|.6.|...|453|
|...|...|627|
+---+---+---+
2 sol.


JPF

[Serg]

Hi, coloin, JPF!
Impressive results! What methods do you use to check - has given pattern valid puzzles or not?

Serg

[JPF]

Code: Select all

 6+---+---+---+        +---+---+---+                                 
  |x..|...|..x|        |9..|...|..7|                                 
  |x..|...|..x|        |3..|87.|..2|                                 
  |x..|...|..x|        |64.|...|..3|                                 
  +---+---+---+        +---+---+---+                                 
  |x..|...|..x|        |46.|...|..8|                                 
  |x..|...|..x|        |2..|.3.|..1|                                 
  |x..|...|..x|        |5..|...|..4|                                 
  +---+---+---+        +---+---+---+                                 
  |x..|...|..x|        |7..|...|..6|                                 
  |x..|...|..x|        |8..|...|..5|                                 
  |xxx|xxx|xxx|        |123|456|789|                                 
  +---+---+---+        +---+---+---+    valid 25plus5 clues  [only one found]   


Here is an other one with symmetry:

Code: Select all

 2 . . | . . . | . . 8
 7 . . | 3 . 9 | . . 6
 5 . . | . . . | . . 1
-------+-------+-------
 6 . . | . 8 . | . . 7
 3 . . | . . . | . . 2
 4 . 9 | . . . | 3 . 5
-------+-------+-------
 8 . . | . . . | . . 4
 9 . . | . . . | . . 3
 1 2 3 | 4 5 6 | 7 8 9


JPF