## minimum number of clues per band/stack

Everything about Sudoku that doesn't fit in one of the other sections

### Re: minimum number of clues per band/stack

dobrichev wrote:
Code: Select all
`3+4+103+5+93+6+83+7+74+4+94+5+84+6+75+5+75+6+6`

I see much work there.

I did not think about feasibility so far, but the couple 3+5+9 and 4+4+9 seems good
If it is feasible, it is likely the limit of what that kind of filter can do
We have puzzles 4+4+9, but not so many and with a very limited number of patterns.

What is for sure is that the filter !(3+4+10) is very efficient in the generation of 17 clues possibly valid patterns.
(killing for example the start 111 111 111)
champagne
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### Re: minimum number of clues per band/stack

coloin wrote:Well no need, assuming there isnt a different double band solvable with 7 clues

Not proved on my side.

JPF
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### Re: minimum number of clues per band/stack

JPF wrote:
coloin wrote:Well no need, assuming there isnt a different double band solvable with 7 clues

Not proved on my side.

JPF

I continue on my side to scan the field. I have a week off, but the work will go on, I hope to have my partial scan covered by the end of the week.
champagne
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### Re: minimum number of clues per band/stack

just before I go these that can be already accounted

12345678945718962368927314524...........9...7.................8...6.4............
12345678945718962368927314524...........9...7.................8...6.2............
12345678945718962368927314524...........3...7.................8...6.4............
12345678945718962368927314524...........3...7.................8...6.2............
champagne
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### Re: minimum number of clues per band/stack

Yes, there are in the list of the 144 puzzles given here.
JPF
JPF
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### Re: minimum number of clues per band/stack

dobrichev wrote:
Code: Select all
`3+4+103+5+93+6+83+7+74+4+94+5+84+6+75+5+75+6+6`

I think we must consider distribution 3+3+11 too, because nobody proved it is impossible for valid puzzles.

Serg
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### Re: minimum number of clues per band/stack

This post is to announce that Mladen (a.k.a. dobrichev) has completed the testing initiated by this post, and the one that preceded it.
The final result was that of the 1,291,535,777,143 puzzles tested, only 4 were valid:

Code: Select all
`1472365892584791363691582745...1...........98.........7...........8.4............1472365892584791363691582745...1...........98.........7...........9.4............147258639259346178368179245.8......6...73.............7.1..............4.........147238695268459137359176284..3.8..........4.6...........5.........6.1............`

By considering replacements for band 1 in those puzzles, the list expands to the 144 (ED) puzzles first mentioned by Colin in this post.
In the list above, the first two puzzles expand to 44 puzzles each, and the next two, to 32 and 24 puzzles respectively.

Congratulations all around, to:

champagne, who found puzzles equivalent to the first 3 above.
coloin, who found a puzzle (or puzzles) equivalent to the 4th one.
dobrichev, who determined that the list of 144 ED 3+4+27 puzzles, is complete
(on the assumption that some guy calling himself blue, hasn't made any mistakes !).
I'll leave it to others, to detail the implications (e.g. no 17 clue puzzle with "3+4+10" form, exists).

P.S. I should mention that Mladen used his 'fsss2' solver here, and it performed wonderfully:
248,000 puzzles per core-second, on a 3.2GHz PC, running 4 threads on 4 cores
284,000 puzzles per core-second, on a 2.7GHz 16-core server, hyperthreaded to 31 threads (I believe).
blue

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### Re: minimum number of clues per band/stack

Serg wrote:I think we must consider distribution 3+3+11 too, because nobody proved it is impossible for valid puzzles.
Serg

Having established that we have no 17 clues puzzles out of a 3+4+27, is it still possible to find a 3+3+27 ? and to have 17 clues puzzles out of it

A question to blue?

Just considering the speed of a brute force with 34 given and the fact that in the process you applied to reduce the count some steps are not so easy (and likely doing part of the job of a brute force), is it really worht to do more than your 842724
IMO, applying directly the 842724 to the 44 till the end (checking validity of puzzles) could be the quickest process

The check of the 3+4+27 has been finally relatively short, what would be the "size factor" for 4+4+27 and 3+5+27. At least for 4+4+27, it seems to me than we are still in the "feasible area"
champagne
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### Re: minimum number of clues per band/stack

champagne wrote:A question to blue?

Just considering the speed of a brute force with 34 given and the fact that in the process you applied to reduce the count some steps are not so easy (and likely doing part of the job of a brute force), is it really worht to do more than your 842724
IMO, applying directly the 842724 to the 44 till the end (checking validity of puzzles) could be the quickest process

It was definitely worth it. For timings on my machine, with 'fsss2' : the initial reduction from 44*842724 to 14760393 <band,pattern> cases, took about 100 seconds, but would save ~4200 hours of CPU time. (The 100s was so small, I didn't bother trying to shorten it). The 2nd reduction (with a "per puzzle" cost), would save ~1050 hours, at a cost of 2 hours. All in all, it gives about a 4:1 reduction, at very little cost.

champagne wrote:The check of the 3+4+27 has been finally relatively short, what would be the "size factor" for 4+4+27 and 3+5+27. At least for 4+4+27, it seems to me than we are still in the "feasible area"

Using the same kinds of reductions, the number of puzzles to be tested, goes from 1.3e+12, to 53.4e+12, based on a small sampling.
They're split pretty evenly between 4+4+27 and 3+5+27 (oddly enough).
With one extra clue, the time to test each puzzle drops to around 81% of what it was for 3+4+27 -- surprise !
The net effect, means around 33.4 times as much CPU time.
With Mladen's 16-core server, it would take ~110 days

I tested around 5.5e+9 puzzles (~a half & half mix), and got 51 results (below) ... 14 3+5+27's and 37 4+4+27's.
Those expanded to 1668 ED puzzles, none of which contained an 18

Hidden Text: Show
3+5+27
1472396852584761393691582744......1..8..67...............3...........4..........6
1472396852584761393691582745......1..8..67...............3...........5..........6
148279563257346189369158427.1..3......4...9.........5...2....7.....1.............
147239856258176394369458127..58..........1.4........6.....6.........7.........9..
147356289268179435359248167..1.....8...6...7......3.......84.........6...........
1472396852691583743584761297..91......3.....6..........2.....1......3............
1472368592684591373591782648.....9...1.3.........6.....7......6...8..............
1472398562581763943694581278.4..........67..5...........6.........3..2...........
147236589268159374359478126..6..4.....5....1........3.4............1.........5...
147236589268159374359478126..6..4.....5....3........1.4............1.........5...
147356289268179435359248167.9.........38.1...........48............9..........3..
147356289268179435359248167..3.........51.6.........4..1............3.9..........
147258639259346187368179245.84.....6...5.............15.....3.......4............
147258639259346187368179245.94.....6...5.............15.....3.......4............

4+4+27
1472563892581394673694781256...........31...4..........1.....5......2.9..........
1472563892581394673694781256...........31...4..........3.....5......2.9..........
1473562892681794353592481678....1......6.............4.9.........5.9...........1.
1472586392593461783681792458....1......6...........9..6............94..7.........
1472365892581793643694581275.1......7.............4....9.....4.....1.2...........
1472365892581793643694581277.1......5.............4....9.....4.....1.2...........
1472563892581397643694781257.1...........5.4...........2.8...........9..........3
1472563892581397643694781257.3...........5.4...........2.6...........9..........3
1472563892581397643694781257.3...........5.4...........2.6...........9..........7
14723658926845913735917826451.3............9...........76............3.........1.
14723658926845913735917826451.3............9...........96............3.........1.
1472398562694581373581762947.1.............6.........9.9.........4.........7..3..
1472398562694581373581762949.1.............6.........5.7.........6.........3..9..
1472586392593461783681792457.......1...6...........8....5..........8..........46.
1472563892581397643694781257.5.........6...4...........8....5.....3...........9..
1472563892581397643694781257.5.........6...4...........8....5.....7...........9..
1472563892581397643694781257.5.........6...4...........8....9.....7...........5..
1472365892584791363691582747.1...........3.........6...9...4..3......7...........
14735826925917643836824915748..........9.5...............41..........8.5.........
147239856258176394369458127.2.....3.....6.........7....3............17.8.........
147239856258176394369458127.2.....3.....8.........4....3............79.8.........
14725689326834915735917842661.............98............2.8..........7..........1
14725689326834915735917842661.............98............2.9..........7..........1
14725689326834915735917842663.............28............2.9..........6..........5
1473582692591764383682491757.1...........5...........4.8....5.....7.........2....
14723865925917638436845912741...........87............8..........36..4...........
1472398562581763943694581275............2...8....4.....86...........15...........
1472398562581763943694581275............2...9....4.....96...........15...........
1472398562581763943694581275............4...9....1.....91...........25...........
1472398562581763943694581275............4...9....2.....96...........15...........
1472398562581763943694581279............6...5....4.....36...........19...........
1472386952684591373591762848.1..........6.4............7..4......2....1..........
14723968526915837435847612971..........9....2...........2.........3..7......1....
1472398562581763943694581275......7....6.4.............84.......1..............3.
1472398562581763943694581275......7....6.4.............86.......1..............3.
1472398562581763943694581275......7....8.1.............86.......2..............3.
1472596832693481573581764297......6....9..........2....9.....3...4..........6....
blue

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### Re: minimum number of clues per band/stack

There are 12998 ed patterns and around 7850 ed patterns after Magic processing.
I tested around 1500 patterns and I got (randomly) 870 valid puzzles before "expansion".
List 1 of 4-4-27.txt
It's where I am.
Edit 1:
I can't say that my puzzles are before "expansion" as defined by blue.
They are, only if, after the minlex transformation of the grid solution, all the 8 "clues" stay in a position > 27

Edit 2: deleted - irrelevant
JPF
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### Re: minimum number of clues per band/stack

JPF wrote:Edit 1:
I can't say that my puzzles are before "expansion" as defined by blue.
They are, only if, after the minlex transformation of the grid solution, all the 8 "clues" stay in a position > 27

From the 871 puzzles, I got 823 ED expansion sets, for a total of 29138 ED puzzles.
The first matchup, was for puzzles 1 and 3:

Code: Select all
`.................1.....2.34....1.......5.......2...6...24.765.358713426.6.3259487 (#1).................1.....2.34....1.......5.......2...6...142785935.71.42.882395641. (#3)`

Filling out band 3, and swapping 8's and 9's in #3, the two puzzles are:

Code: Select all
`. . . | . . . | . . .. . . | . . . | . . 1. . . | . . 2 | . 3 4------+-------+------. . . | . 1 . | . . .. . . | 5 . . | . . .. . 2 | . . . | 6 . .------+-------+------6 1 4 | 2 7 8 | 5 9 35 9 7 | 1 3 4 | 2 6 88 2 3 | 9 5 6 | 4 1 7. . . | . . . | . . .. . . | . . . | . . 1. . . | . . 2 | . 3 4------+-------+------. . . | . 1 . | . . .. . . | 5 . . | . . .. . 2 | . . . | 6 . .------+-------+------8 2 4 | 9 7 6 | 5 1 35 9 7 | 1 3 4 | 2 6 86 1 3 | 2 5 8 | 4 9 7`

Band 3 in those puzzles, has the same digits in each of the 9 mini-columns ==> they're in the same gangster class.
The tops match too ==> they have the same expansion sets.

There were a lot of 18's here ... 69 ... and no 17's.

Hidden Text: Show
.................1..2.34.................2...15......6..4.5.9....8.6..3..7.8..24.
................12....34.................1..4..5...6..1.....8..4..25.9..8.3.6..5.
................12....34.................1..5..6...3..95.2..8...1...54....4.6.7..
................12....34.................1..5..6...3....4.6.8...1...54..85.2..7..
................12....34.................1.5...6...3..2...6...9.3.2..7..71..9.4..
................12..3..4.........3......1.....2..5.....5..8.....17..94...6...28.3
..............1..2..2....3...............45....3....6.85.93.1......6.9...4.2....8
..............1..2..2....3...............45....3....6..8..6.1.....2....76.593.8..
..............1..2..2....3...............45....3....6.4..2....8....6.9..87.93.1..
........1.......2...3..4.......1........5...6.2.......5.4..6..96...7.2..9....8.4.
........1.......23.....4.......2......5........6...7...2.1....53....68...1..974..
........1.......23.....4.......2......5........6...7...1..584..3....68..52.1.....
........1.......23.....4.......2......5........6...7...1..584..3....68.5.2.1.....
........1.......23.....4.......2......5........6...7...1..584..37...68...2.1.....
........1.......23.....4.......3......4...5....6......13..2..5..2...7.6..9..1.4..
........1.......23.....4.......3......5........6...5...1289..3..8...64...3..1....
........1.......23.....4.......3......5........6...5...1289.7...8...64...3..1....
........1.......23.....4.......3......5........6...5..4.28.57...8...64...3..1....
........1.......23.....4.......3......5........6...5..41..957...8...64...3..1....
........1.......23.....4.......3......5........6...5..41.89.7...8...64...3..1....
........1.......23.....4.......3......5........6...7...1..865..3..1...9..2...74..
........1.......23.....4.......3......5........6...7..3...6...81...2.4...2...765.
........1.......23.....4.......3......5........6...7..3...1..5.12.8..4..8....76..
........1.......23.....4.......3......5........6...7..3...6...81...2.4...2...965.
........1.......23.....4.......3......5........6...7..3...1..5.12.8..4..8....96..
........1.......23.....4.......3......5........6...7..3...2.4..1...6...8.2...965.
........1.......23.....4.......3......5........6...7..3...2.4..1...6...8.2...765.
........1.....2........3.4........2...1........5.6....9....5..32...9..6.34..8.7..
........1.....2........3.4........2...1.5......6.......3..1...827...4.3...9.6...5
........1.....2........3.4........2...5........6.7....42.9..3...3..8...6.9..1...7
........1.....2........3.4........2...5........6.7....42.9..3...3.2....6.9..1...7
........1.....2........3.4........2...5........6.7....42.9..3...3..1...6.9..8...7
........1.....2........3.4........2...5........6.7....42.9..3...3.2....6.9..8...7
........1.....2........3.4........2...5........6.7....42.9..3...3..8...6.9..1...8
........1.....2........3.4........2...5........6.7....42.9..3...3.2....6.9..1...8
........1.....2.......3..4.........5....1......6...7....7..6.2..1.85.3...5..4.9..
........1.....2.....3.....4.......5.....13....6........8..4...7...1...2.9.467..8.
........1.....2.....3.....4.......5...6........7.1....9...5..6..2...6..75..4.9.2.
........1.....2.....3.....4.......5...6........7.4....2....85...5.3.9.7..9..6..8.
........1.....2.....3.....4.......5...6........7.4....2....756..5..1..3..8.6...9.
........1.....2.....3.....4.......5...6........7.4....2....856..5..1..3..8.6...9.
........1.....2.....3.....4....1.....2.....5..6.........1.4...3.87..5.6.9....6.2.
........1.....2.....3.....4....1.....2.....5..6.........1..7.9...4..8..69..2.6.7.
........1.....2.....3.....4....1.....2.....5..6.........4..8..2..1..7.9.9..2.6.7.
........1.....2.....3.....4....4.....2.....5..6.........7..5.23..5..9.6.3.9.1....
........1.....2.....3.....4....4.....2.....5..6.........7..592...5..9.6.3.9.1....
........1.....2.....3.....4....4.....2.....5..6.........7..592...5..9.6.38..1....
........1.....2.....3.....4....4.....2.....5..6.........7..592...931.....4..7..6.
........1.....2.....3.....4....4.....2.....5..6.........5..9.2...7..5.633.9.1....
........1.....2.....3.....4....4.....2.....5..6.........5..9.2...7..596.3.9.1....
........1.....2.....3.....4....4.....2.....5..6.........5..9.2...7..596.38..1....
........1.....2.....3.....4....4.....2.....5..6.........5..9.2...938.....4..1..68
........1.....2.....3.....4....4.....2.....5..6.........1..8..3...5.6.7.9.4.3..6.
........1.....2.....3.....4....4.....2.....5..6.........1..8..3...5.6.7.9.4.3.1..
........1.....2.....3.....4....4.....2.....5..6.........1..8..3...5.7.6.9.4.3..7.
........1.....2.....3.....4....4.....2.....5..6.........1.3..2..5...7.6.9.4.1...8
........1.....2.....3.....4....4.....2.....5..6.........1.9..2......657.9.4.3..6.
........1.....2.....3.....4....4.....2.....5..6.........1.9..2......657.9.4.3.1..
........1.....2.....3.....4....4.....2.....5..6.........1.9..2......756.9.4.3..7.
........1.....2.....3.....4....4.....2.....5..6.........4.9..2......657.9.1.3..6.
........1.....2.....3.....4....4.....2.....5..6.........4.9..2......657.9.1.3.4..
........1.....2.....3.....4....4.....2.....5..6.........4.9..2......756.9.1.3..7.
........1.....2.....3.....4....4.....2.....5..6.........4.9..7.....1..6.9.143...8
........1.....2.....3.....4....4.....2.....5..6.........4.9..7....5.7.6.9.1.3...8
........1.....2.....3.....4....4.....2.....5..6.........4.9.1......1..6.97.43...8
........1.....2.....3.....4....4.....2.....5..6..........9.6.8.3.4.7..6...1..8..7
........1.....2.....3.....4....4.....2.....5.6............1.7....183..6.56.9...2.
........1.....2.....3....4.........5.....62....4.......9..4..6..5..8..1.62..3.8..
........1.....2.....3....4.......2....5.1.....6.........4..3...29.4..61..7..6.9..
blue

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### Re: minimum number of clues per band/stack

In some ways the exercise was a double success.
Not only were 3+4+27 puzzles found - but all of them were found.
This means that in any search the 3+4 exclusion can effectively be employed as a filter.

However,unless we get further {unknown} reductions the 3+5 and 4+4 cases are beyond the above spectacular efforts.
Maybe useful information will come from 3+5 and 4+4 patterns shown not to have puzzles ....this could then be employed as a further filter.
Maybe the Magic pattern can be extended to cover two bands [I think Serg just looked at single bands [B1B2B3] and crossing bands [B1B2B3B4B7].

As a start, this valid puzzle demonstrates the need for at least 4 clues in B1B2B3B4B6, and fairly rare. [Not proven , I hasten to add.....]
Code: Select all
`+---+---+---+|.1.|...|...||...|...|.24||...|...|...|+---+---+---+|4..|123|...||...|456|...||...|789|...|+---+---+---+|296|538|147||574|261|983||831|947|652|+---+---+---+`

Even adding 5 clues its sometimes difficult to get a puzzle....
For example, I very much doubt whether the box distribution has valid puzzles ?
Code: Select all
`041140999`

Edit
.... maybe not as this puzzle is
Code: Select all
`130051999+---+---+---+|1..|2..|...||...|.34|...||...|...|...|+---+---+---+|...|15.|...||...|627|...||...|...|..8|+---+---+---+|638|792|154||794|815|263||521|463|879|+---+---+---+`
C
Last edited by coloin on Mon Nov 17, 2014 8:56 am, edited 2 times in total.
coloin

Posts: 1877
Joined: 05 May 2005

### Re: minimum number of clues per band/stack

Well it seems Serg and eleven did the work on this.... here

The thread confirmed that the 22 ED [36*] ways of this pattern did not have valid puzzles - and we know that now
Code: Select all
`111111999`

The thread confirmed that puzzles with this pattern all had valid puzzles [ all 20 ED patterns] [*Excluding 2 empty rows]
Code: Select all
`111119999`

However eleven generated all the ways for valid puzzles of that pattern and none could be reduced from 9 to 2 clues in B6 - and we know this now too
Code: Select all
`111112999`

We have since proved that
Code: Select all
`011   011323   332332   332`
is a required distribution if 2 clues in a band - so all puzzles with 2 clues in a band can be essentially ignored.

Patterns with an empty box were outside the scope of the thread ..... but this one did have valid puzzles
Code: Select all
`111019999`

C
coloin

Posts: 1877
Joined: 05 May 2005

### Re: minimum number of clues per band/stack

blue wrote:From the 871 puzzles, I got 823 ED expansion sets, for a total of 29138 ED puzzles.
...
There were a lot of 18's here ... 69 ... and no 17's.

Thanks for analyzing those data.
I produced more puzzles for a while and got 135 18's that are not in your list. -Thanks to Mladen's checker.
Here they are:
18-clue puzzles.txt
Edit: Here are 242 more:
Hidden Text: Show
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I got one 17, but unfortunately not "new". It's one of the 17 known clue puzzles with a 4-4-9 distribution.

JPF

PS: checker is still running, producing around 15 new 18's per hour...Edit: I stopped it!
JPF
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Posts: 4022
Joined: 06 December 2005
Location: Paris, France

### Re: minimum number of clues per band/stack

For champagne.

(ED 4+4+27 and 3+5+27 patterns, after magic 40 filtering)

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Posts: 894
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