## One-clue-boxes patterns

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

### Re: One-clue-boxes patterns

Hi, coloin!
coloin wrote:probably all the possible
Code: Select all
`111191999`
have puzzles ?

Maybe it would be possible to generate all the possible
Code: Select all
`1111-1`
B12346 combinations which are solveable with 5 [one in each box] clues - are there really that many ?

During my search I was really interested in principle question - do the map examined have valid puzzles at all? So, I've done not exhaustive, but partial search for valid puzzles having the map
Code: Select all
`1 1 11 1 99 9 9`

It turned out that some patterns had valid puzzles even during partial search. It was sufficient to say that this map has valid puzzles. But I agree, it makes sense to study every pattern having this map to determine - does it have any valid puzzles? If there are some patterns which have no valid puzzles, it would be useful for further filtering out invalid symmetric 18-clue patterns (at least). I'll do such search in the coming days.
coloin wrote:If there are not to much which case we could easily resolve the next level
Code: Select all
`111121999`

We know the map
Code: Select all
`1 1 11 1 99 9 9`

has valid puzzles, but the map
Code: Select all
`1 1 11 1 19 9 9`

has no valid puzzles. If we will increase number of clues in B6 box (2,3, ... 7 clues), we'll found "boundary" map having no valid puzzles with maximal number of clues in B6 box. More clues will be in B6 box for boundary map, more useful will be such "exclusive" map. Even intermediate results - knowing patterns not having valid puzzles in the case when its map have valid puzzles (other patterns of this map have valid puzzles) - will be useful for invalid patterns exclusion.
Maybe I'll do it in the coming days.

Serg
Serg
2018 Supporter

Posts: 606
Joined: 01 June 2010
Location: Russia

### Re: One-clue-boxes patterns

Well it seems all the potential patterns with 5 one-clue-boxes
Code: Select all
`111191999`

have puzzles
Code: Select all
`1...........2...........3...4.816.5....925......734...9723615846154782934835927161...........2...........3...4.538......129.5....674...9518437267349625812867514391...........2...........3..4..981.5....753......642...8394267157243158965168792431...........2...........3..4..8375.....529......461...3941867526527438918719524361...........2...........3..4..613......428.5....579...9367421857158369428421957361.....2.....3...............4.281......673.5....594...2758461394831596726197324851.....2.....3...............4.276.5....483......159...6789421354318659722597316841.....2.....3..............4..123......649.5....587...6298157437139648258457321691.....2.....3..............4..751......6245.....893...5149367822391876546875421931.....2.....3..............4..821.5....574......963...7346521895184976232691384751.....2.....3..............4..8625.....439......517...543726189689143725721958643                                                                                 1..2...........3............4.7935.....642......581...8651794322914368574378259611..2...........3............4.386.5....791......542...8571349622319678459648257311..2...........3............4.589......127.5....634...4628537197859164239317426851..2...........3...........4..6785.....359......124...8347956122794618536518327941..2...........3...........4..182.5....549......376...6849157235314279682798635411..2...........3...........4..123......467.5....859...615932784948571263732684591`

this is the reciprical complete B12346 in the first puzzle.
Code: Select all
`134657829598243671726189345347...952861...437259...168...........................`

will just try to find a "few" more....

edit - added the ones i forgot !
C
coloin

Posts: 1727
Joined: 05 May 2005

### Re: One-clue-boxes patterns

this all implies that there arnt puzzles with
Code: Select all
`101     111     011191     091     191999     999     999`

C
coloin

Posts: 1727
Joined: 05 May 2005

### Re: One-clue-boxes patterns

Serg wrote:I would take about 15 minutes per all stages because there are 20 valid B1+B2+B3+B4+B5 one-clue-box configurations only.

I dont understand that. Could you post the 20 please and explain more ?
eleven

Posts: 1847
Joined: 10 February 2008

### Re: One-clue-boxes patterns

Hi, eleven!
eleven wrote:
Serg wrote:I would take about 15 minutes per all stages because there are 20 valid B1+B2+B3+B4+B5 one-clue-box configurations only.

I dont understand that. Could you post the 20 please and explain more ?

This is wrong statement. I'll edit that post.

Serg
Serg
2018 Supporter

Posts: 606
Joined: 01 June 2010
Location: Russia

### Re: One-clue-boxes patterns

Serg wrote:I would take about 15 minutes per all stages because there are 20 valid B1+B2+B3+B4+B5 one-clue-box configurations only.

if it helps there are the 17 above along with these 6 multisol invalids [two empty rows in a band]
Code: Select all
`1..3..2.....................4.281......673.5....594...2758461394831596726197324851..3..2.....................4.276.5....483......159...6789421354318659722597316841..3..2....................4..123......649.5....587...6298157437139648258457321691..3..2....................4..751......6245.....893...5149367822391876546875421931..3..2....................4..821.5....574......963...7346521895184976232691384751..3..2....................4..8625.....439......517...543726189689143725721958643  [all multisol.]`

if i have havent missed 3 others somewhere.....
C
coloin

Posts: 1727
Joined: 05 May 2005

### Re: One-clue-boxes patterns

There are thousands unique. From the 3 best populated 5 box sets i did not find any with 2 clues in the 6th box, but some with 3:

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

Code: Select all
`1..2...........3...........3.....1.7....6..9..........2197845635431268797863952411..2...........3...........5.......1....6.9.........7.2958764134615932878731246951..2...........3...........2......91....7.5...........4561827398729631549314578261..2...........3...........2......91....7.8...........4321675898769531249514827361..2...........3...........5.......1....6.9.........8.4957236187615482938231964751..2...........3...........7.....8.1....6...........7.2978451635431962878617235941..2...........3...........9......24....7.5...........2769834514591678328314527961..2...........3...........9......24....7.8...........2769534814591827368314675921..2...........3...........9......42....7.8...........2591674383814527966749385211..2...........3...........8......24....7.9...........2981574363548627196714395821..2...........3...........8......24....6.7...........4831596725298764317164329851..2...........3...........8......24....6.9...........4231796855864327917198564321..2...........3...........9......42....7.6...........4391875266514327898729654131..2...........3...........9......42....7.5...........479185623651432789832967415`

I dont have time now to try all with 2, but it could be done in a day (more time for adopting the code than for cpu).
eleven

Posts: 1847
Joined: 10 February 2008

### Re: One-clue-boxes patterns

Hi, colleagues!
I've done exhaustive search for all patterns having map
Code: Select all
`1 1 11 1 99 9 9`

If am not mistaken there exist 20 essentially different such patterns. It turns out that each pattern has valid puzzles. Here are examples:
Code: Select all
`        P1                      P2                      P3                      P41 . . 5 . . . . .       1 . . 5 . . . . .       1 . . 5 . . . . .       1 . . 5 . . . . . . . . . . . 3 . .       . . . . . . 3 . .       . . . . . . 7 . .       . . . . . . 7 . . . . . . . . . . .       . . . . . . . . .       . . . . . . . . .       . . . . . . . . . 2 . . 6 . . 1 5 9       2 . . . 3 . 6 9 1       . 4 . . 3 . 1 5 9       2 . . . . . 1 6 9 . . . . . . 4 8 6       . . . . . . 5 2 8       . . . . . . 2 6 4       . . . 6 . . 5 8 2 . . . . . . 7 2 3       . . . . . . 7 3 4       . . . . . . 3 7 8       . . . . . . 3 7 4 6 1 2 8 5 4 9 3 7       5 1 2 7 6 9 8 4 3       6 1 4 2 5 3 9 8 7       6 1 2 9 3 4 8 5 7 8 3 4 9 7 6 5 1 2       6 3 8 1 2 4 9 7 5       8 3 2 6 9 7 4 1 5       8 7 4 1 6 5 9 2 3 9 7 5 1 2 3 8 6 4       9 4 7 8 5 3 1 6 2       9 7 5 8 1 4 6 2 3       9 3 5 2 8 7 4 1 6         P5                      P6                      P7                      P81 . . 5 . . . . .       1 . . 5 . . . . .       1 . . . . . . . .       1 . . . . . . . . . . . . . . 7 . .       . . . . . . 3 . .       . . . 3 . . 7 . .       . . . 7 . . 8 . . . . . . . . . . .       . . . . . . . . .       . . . . . . . . .       . . . . . . . . . 2 . . . . . 1 5 9       . 4 . . . . 5 3 7       2 . . 5 . . 3 9 4       2 . . . 4 . 9 1 6 . . . . 4 . 3 2 7       . . . . 8 . 2 4 9       . . . . . . 6 5 8       . . . . . . 3 8 4 . . . . . . 4 8 6       . . . . . . 1 8 6       . . . . . . 2 1 7       . . . . . . 2 5 7 6 1 2 9 5 3 8 7 4       6 1 2 8 3 5 7 9 4       6 3 2 8 1 4 9 7 5       3 1 5 6 8 4 7 9 2 8 3 5 1 7 4 9 6 2       8 7 4 1 2 9 6 5 3       8 7 1 9 5 2 4 6 3       6 7 2 5 1 9 4 3 8 9 7 4 6 2 8 5 1 3       9 3 5 4 6 7 8 1 2       9 4 5 7 6 3 8 2 1       9 4 8 2 3 7 5 6 1         P9                      P10                     P11                     P121 . . . . . . . .       1 . . . . . . . .       1 . . . . . . . .       1 . . . . . . . . . . . 7 . . 8 . .       . . . 2 . . 8 . .       . . . 7 . . 8 . .       . . . 8 . . 2 . . . . . . . . . . .       . . . . . . . . .       . . . . . . . . .       . . . . . . . . . . 6 . 4 . . 3 9 1       . 3 . . 1 . 9 4 8       2 . . . . . 4 5 9       2 . . . . . 5 9 8 . . . . . . 2 8 4       . . . . . . 2 3 7       . . . 9 . . 3 7 8       . . . . 5 . 7 1 6 . . . . . . 5 7 6       . . . . . . 1 6 5       . . . . . . 2 6 1       . . . . . . 3 4 2 3 1 2 6 7 4 9 5 8       3 1 2 8 4 5 7 9 6       3 1 2 4 9 6 7 8 5       3 1 2 5 4 8 9 6 7 6 4 5 8 9 2 7 1 3       6 4 8 9 3 7 5 1 2       6 4 5 8 1 7 9 2 3       6 4 5 7 3 9 8 2 1 9 7 8 3 1 5 4 6 2       9 7 5 1 6 2 3 8 4       9 7 8 3 5 2 6 1 4       9 7 8 2 1 6 4 3 5         P13                     P14                     P15                     P161 . . . . . . . .       1 . . . . . . . .       1 . . . . . . . .       1 . . . . . . . . . . . 8 . . 7 . .       . . . 8 . . 2 . .       . . . 3 . . . . .       . . . 9 . . . . . . . . . . . . . .       . . . . . . . . .       . . . . . . 4 . .       . . . . . . 4 . . . 3 . . . . 4 9 1       . 3 . . . . 4 1 6       2 . . 7 . . 3 9 1       2 . . . 5 . 6 8 9 . . . 2 . . 3 7 8       . . . . 2 . 7 9 8       . . . . . . 2 4 8       . . . . . . 1 3 4 . . . . . . 2 5 6       . . . . . . 3 5 2       . . . . . . 6 7 5       . . . . . . 2 5 7 3 1 2 5 4 8 9 6 7       3 1 2 5 8 6 9 7 4       3 1 2 9 6 7 8 5 4       3 1 2 8 9 5 7 4 6 6 7 5 9 2 1 8 3 4       6 4 5 7 9 1 8 2 3       6 7 5 1 8 4 9 3 2       6 4 5 3 1 7 8 9 2 9 4 8 7 3 6 5 1 2       9 7 8 2 4 3 5 6 1       9 4 8 2 3 5 7 1 6       9 7 8 6 2 4 5 1 3         P17                     P18                     P19                     P201 . . . . . . . .       1 . . . . . . . .       1 . . . . . . . .       1 . . . . . . . . . . . 8 . . . . .       . . . 8 . . . . .       . . . 9 . . . . .       . . . 9 . . . . . . . . . . . 6 . .       . . . . . . 3 . .       . . . . . . 4 . .       . . . . . . 5 . . . 3 . . 6 . 4 9 1       2 . . . . . 1 5 6       2 . . . . . 6 8 7       . 3 . . . . 6 1 8 . . . . . . 3 7 8       . . . 7 . . 4 9 8       . . . . 4 . 1 3 9       . . . . 4 . 3 5 7 . . . . . . 2 6 5       . . . . . . 2 3 7       . . . . . . 2 4 5       . . . . . . 2 9 4 3 1 2 9 7 4 5 8 6       3 1 2 6 5 8 9 7 4       3 1 2 7 9 5 8 6 4       3 1 2 6 7 4 9 8 5 6 4 8 1 3 5 9 2 7       6 4 8 3 7 9 5 1 2       6 4 5 2 1 8 7 9 3       6 4 8 1 9 5 7 2 3 9 7 5 2 8 6 1 4 3       9 7 5 4 2 1 6 8 3       9 7 8 6 3 4 5 1 2       9 7 5 8 3 2 4 6 1 `

The same in linear form:
Code: Select all
`1..5...........3...........2..6..159......486......7236128549378349765129751238641..5...........3...........2...3.691......528......7345127698436381249759478531621..5...........7............4..3.159......264......3786142539878326974159758146231..5...........7...........2.....169...6..582......3746129348578741659239352874161..5...........7...........2.....159....4.327......4866129538748351749629746285131..5...........3............4....537....8.249......1866128357948741296539354678121...........3..7...........2..5..394......658......2176328149758719524639457638211...........7..8...........2...4.916......384......2573156847926725194389482375611...........7..8............6.4..391......284......5763126749586458927139783154621...........2..8............3..1.948......237......1653128457966489375129751623841...........7..8...........2.....459...9..378......2613124967856458179239783526141...........8..2...........2.....598....5.716......3423125489676457398219782164351...........8..7............3....491...2..378......2563125489676759218349487365121...........8..2............3....416....2.798......3523125869746457918239782435611...........3...........4..2..7..391......248......6753129678546751849329482357161...........9...........4..2...5.689......134......2573128957466453178929786245131...........8...........6...3..6.491......378......2653129745866481359279752861431...........8...........3..2.....156...7..498......2373126589746483795129754216831...........9...........4..2.....687....4.139......2453127958646452187939786345121...........9...........5...3....618....4.357......294312674985648195723975832461`

You can see that patterns P11, P12, P18 are new for coloin's list.

The next my goal - to check, do the patterns having map
Code: Select all
`1 1 11 1 79 9 9`

have valid puzzles? But I need a program determining all e-d patterns for given map. These patterns hardly can be found manually because they are several hundred. So, I can do it in some time.

Serg
Serg
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Posts: 606
Joined: 01 June 2010
Location: Russia

### Re: One-clue-boxes patterns

Hi, coloin!
coloin wrote:this all implies that there arnt puzzles with
Code: Select all
`101     111     011191     091     191999     999     999`

Why have you decided that all these 3 maps have no valid puzzles?

Serg
Serg
2018 Supporter

Posts: 606
Joined: 01 June 2010
Location: Russia

### Re: One-clue-boxes patterns

Now i have searched through all possible 6-box-completions for puzzles with 1 given in the first 5 boxes. There was none, which had a pair in the 6th box, which together with the 5 cells could fix all UAs in the first two bands.

So, if i made no mistake, this pattern is invalid:
Code: Select all
`1 1 11 1 29 9 9`
eleven

Posts: 1847
Joined: 10 February 2008

### Re: One-clue-boxes patterns

Hi, eleven!
eleven wrote:So, if i made no mistake, this pattern is invalid:
Code: Select all
`1 1 11 1 29 9 9`

Very interesting result! So, we could filter out new portion of symmetric 18-clue puzzles. How many patterns did you process?
2 clues in B6 box seems to me maximal, because you have already published valid puzzle having 3 clues in B6 box.

Serg
Serg
2018 Supporter

Posts: 606
Joined: 01 June 2010
Location: Russia

### Re: One-clue-boxes patterns

Hi Serg,

this list shows, how much completions for the 5 boxes i processed (pattern/count).
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`1..1...........1...........1..1..................................................      798 1..1...........1...........1...1.................................................     2520 1..1...........1...........1...........1.........................................      546 1..1...........1...........1............1........................................    17506 1..1...........1............1..1.................................................    27402 1..1...........1............1...........1........................................    24723 1...........1...........1..1..1..................................................      870 1...........1...........1..1...1.................................................    21362 1...........1...........1..1...........1.........................................     2764 1...........1...........1..1............1........................................    169061...........1...........1...1..1.................................................    22796 1...........1...........1...1...........1........................................    46237 1...........1..1...........1..1..................................................     1236 1...........1..1...........1...1.................................................    61728 1...........1..1...........1...........1.........................................    25271 1...........1..1...........1............1........................................    36033 1...........1..1............1.1..................................................    11828 1...........1..1............1..1.................................................    42158 1...........1..1............1..........1.........................................    46777 1...........1..1............1...........1........................................   221151                                                                            total   630612`

Note that these sets are not e-d. I have used the column/stack and band 1 automorphisms of the 5 clue pattern only. Especially i did not use row 56 swappaing (if available), but i did use column 89 swapping.
Then each of these completions leaves 8 possible fillings for box 6 (only). So i simply tested all the 36 ways to take 2 cells in box 6 and checked the UA's in bands 12.

It would be nice, when you could validate this result. With all the case distinctions and consecutive steps (band 1, then 4,5,6 boxes) there is a big potential for bugs. I had made some checks (and also found bugs), but not enough to be sure, that all is correct.
[Edit:] corrected the sum
eleven

Posts: 1847
Joined: 10 February 2008

### Re: One-clue-boxes patterns

Thanks for spotting the 3 that i missed.......

I suppose the assumption that these patterns are invalid
Code: Select all
`101     111     011191     091     191999     999     999`

is based on the fact that my generator drew a blank after half an hour with them. i made a lot of puzzle and didnt find a single superfluos clue in the 5 boxes ..................
i did find the
Code: Select all
`102190999`
though

Well done eleven on that. I had hoped it could be proved.
More B12345 solvable with 5 one-in-box clues than i thought, im not sure how you did it so easily !!
i had noted that there could have been some B12345 combinations with several different 5 one-in-box completions.

so , if verified, there is no puzzle with 3,4,27 [11] clues in 3 bands [edit eg with 111,112,344]

with 2 clues in the first band only [cant be 2 in a box] only one band out of the 416 has this property
Code: Select all
` 110223999 `

Code: Select all
`+---+---+---+|1..|...|...||...|2..|...||...|...|...|+---+---+---+|.2.|.3.|.4.||..5|.1.|.7.||...|...|..6|+---+---+---+|492|381|765||357|649|821||681|725|439|+---+---+---+`

we might be able to go one better than this valid combination - but somehow its not that easy.
Maybe a
Code: Select all
`110129999`
is possible.....
C
coloin

Posts: 1727
Joined: 05 May 2005

### Re: One-clue-boxes patterns

coloin wrote:
Code: Select all
`101     111     011191     091     191999     999     999`

If my 1-in-5-boxes sets (which should be supersets for the 5-box completions of these patterns) are complete, only the second of them has unique puzzles:
Code: Select all
` +-------+-------+-------+ | 1 . . | . . . | . . . | | . . . | 2 . . | 3 . . | | . . . | . . . | . . . | +-------+-------+-------+ | . . . | 3 . . | 2 7 5 | | . . . | . . . | 4 8 1 | | . . . | . . . | 6 3 9 | +-------+-------+-------+ | 7 4 6 | 5 3 9 | 8 1 2 | | 8 1 3 | 7 4 2 | 5 9 6 | | 9 5 2 | 6 8 1 | 7 4 3 | +-------+-------+-------+1...........2..3..............6..275......481......6397463815928125697439537428161...........2..3.................283....4.619......547489312765523764891671895432`
eleven

Posts: 1847
Joined: 10 February 2008

### Re: One-clue-boxes patterns

eleven that is a rare one

with only 2 clues in the first band this seems to preclude a puzzle with a one-clue-box in any of the other boxes
even
Code: Select all
`110       011199  or   199999       999`
doesnt come up with a valid puzzle

so the best might well be
Code: Select all
`110223999 but not 110222999`

C
Last edited by coloin on Tue May 15, 2012 2:58 pm, edited 1 time in total.
coloin

Posts: 1727
Joined: 05 May 2005

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