## Sudoku space

Everything about Sudoku that doesn't fit in one of the other sections
I think you have only 34 in your list [1+8,1+9 missing]

thanks for pointing that out. i wrote it out, didin't check them all. was just siting an example. ill correct it

36 is the correct number.

126? i'll have to check my lists again. the mathmatical number is correct. i didn't verify it using the formual you posted. i only did a manual tally on paper and got 125. i'll double check it.

i am using quad as in a quadrant, mostly i see it refered to as a 3*3 square or a box on here. to me there are 9 quads/boxes/

Code:
basic evaluation of limitations noted above (9+12) to to each successive clue.

Is this the same as the pencil mark reductions by a single clue ?
This is maximally 9-within a 9x9 box, and 2x6 -for the line interactions in the horizontal and vertical chutes.

correct - this would be penicl mark reductions of each individual clue - reducing the aviable subgrid cells for each subsequent selections.

problem i found using that method of assinng a math formual derived from line of sites over lapping and being counted mutiple times.

Code: Select all
` 3 top row example. *-----------* |1..|...|...| |...|1..|...| |...|...|1..| |---+---+---| |...|...|...| |...|...|...| |...|...|...| |---+---+---| |...|...|...| |...|...|...| |...|...|...| *-----------*`

say i map out three clues over the top three boxes it should reduce the

avialable space by 9*3 + 6*3 = 45 leaving the 4th clue with (81-45)= 36 vailid spaces to pic from.

if i did it the intal way above id get a sumation error due to overlaping line of sights. (81-(9*3)+((6*2)*3) = 18 remaining for the 4th clue.

so i ploted out a sumation correction correlation based on line of sight for each combination of cells for the single digit. which wil add back on the extra reduction on to the total number of valid cells remaing.

So if you have 16 clues you just cant do what we know we can do with 17 clues.........
So, how does 17 clues manage it then !!! .

my theory is the existance of conflict of placement. each number must compelt all 9 subgrids to express a unique solution.
to do so, each heuristic or rookeir. overlaps creating "locked" candidates into specific cells. definiing each other heuristic to either complet further and reduce other rookeries to completed or near completed stages.

ie
a rookeri pattern of
Code: Select all
` *-----------* |1..|...|...| |...|1..|...| |...|...|1..| |---+---+---| |.1.|...|...| |...|.1.|...| |...|...|.1.| |---+---+---| |..1|...|...| |...|..1|...| |...|...|..1| *-----------*`

will remove every single instance of these placemetns from all rookeries. of the number 2-9. dramtically reducing the number of vaild rookeries for each other number sets. in some cases perhaps enough to lock a rookeir to compelt to further point of singles or complet fully.

so if u reduce subgrids via rookeri patterns. u will leave other rookeris with 1 pattern left to use.
or mutiple valid patterns(2+ valid rookes same space) ie no pattern reduction in subgrids where 2 or more choices can be vaild (mutiple solutions).

or no valid pattern = conflict of placement no vaild placement for any rookeri(incorect placemnt, or simple no vaild pattern completions as it cannot complet to 9 digits placed. can arrive where 2 numbers must be vaild at the same time same space.

my theory is that 17 numbers creates enough overlapping heuristics that all singles are expressed and valid in each subgrid.

and that a 16 set of right cominations of heuristics can reduce to only a min of 2 uncompleted rookiers with 2 distinctive possible completions. either of the 2 spaces can be either number.

id post an example from one of the many known 16 grids where there is perciesly 2 rookeris on the same space, where they are the only uncompleted single spaces all with exactly 2 choices. but i am lacking an example atm.

menaing that the existiance of the 17th number defines the unbalanced rookeire and forces the cells to choose a single path rahter then bifucating.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

thanks red ed,
thanks StrmCkr...I can see the way you are going, and i understand the way the rookeries will and wont fit !

with respect to "line -of-sight" eliminations

Code: Select all
`and that a 16 set of right cominations of heuristics can reduce to only a min of 2 uncompleted rookiers with 2 distinctive possible completions. either of the 2 spaces can be either number`

The infamous 16-clue subpuzzle [with 2 solutions] [18 clues unavoidable set][a 2-rookery][from the SF grid][2-perm]
Code: Select all
`+---+---+---+|...|.4.|7..||.8.|...|...||.1.|...|.2.|+---+---+---+|...|8..|..6||7..|...|...||4..|...|2..|+---+---+---+|3.2|.7.|...||...|...|...||...|..6|.18|+---+---+---+Here are the initial [virtual] pencilmarks, reduced by the "in-line" reductions, 16 numbers plus pencil marks+-------------------------+-------------------------+-------------------------+| 2569    23569   3569    | 123569  4       123589  | 7       35689   1359    | | 2569    8       345679  | 1235679 123569  123579  | 134569  34569   13459   | | 569     1       345679  | 35679   35689   35789   | 345689  2       3459    | +-------------------------+-------------------------+-------------------------+| 1259    2359    1359    | 8       12359   1234579 | 13459   34579   6       | | 7       23569   135689  | 1234569 123569  123459  | 134589  34589   13459   | | 4       3569    135689  | 135679  13569   13579   | 2       35789   13579   | +-------------------------+-------------------------+-------------------------+| 3       4569    2       | 1459    7       14589   | 4569    4569    459     | | 15689   45679   1456789 | 123459  123589  1234589 | 34569   345679  234579  | | 59      4579    4579    | 23459   2359    6       | 3459    1       8       | +-------------------------+-------------------------+-------------------------+Here are the almost final stages with simple techniques, most solvers stop here, SE says needs nishio or forcing chains to show 5 or 9 cannot be at r4c3 !+-------------------------+-------------------------+-------------------------+| 6       359     359     | 2       4       1       | 7       8       59      | | 2       8       47      | 579     6       359     | 1       59      34      | | 59      1       47      | 579     8       359     | 6       2       34      | +-------------------------+-------------------------+-------------------------+| 1       2       359     | 8       359     7       | 59      4       6       | | 7       359     6       | 4       359     2       | 8       359     1       | | 4       359     8       | 6       1       59      | 2       359     7       | +-------------------------+-------------------------+-------------------------+| 3       4       2       | 1       7       8       | 59      6       59      | | 8       6       1       | 59      59      4       | 3       7       2       | | 59      7       59      | 3       2       6       | 4       1       8       | +-------------------------+-------------------------+-------------------------+This is the "real" pm board which show the 2-digit unavoidable,16 given clues, plus 47 inserted clues, plus 36pm+-------------------------+-------------------------+-------------------------+| 6       3       59      | 2       4       1       | 7       8       59      | | 2       8       4       | 7       6       59      | 1       59      3       | | 59      1       7       | 59      8       3       | 6       2       4       | +-------------------------+-------------------------+-------------------------+| 1       2       3       | 8       59      7       | 59      4       6       | | 7       59      6       | 4       3       2       | 8       59      1       | | 4       59      8       | 6       1       59      | 2       3       7       | +-------------------------+-------------------------+-------------------------+| 3       4       2       | 1       7       8       | 59      6       59      | | 8       6       1       | 59      59      4       | 3       7       2       | | 59      7       59      | 3       2       6       | 4       1       8       | +-------------------------+-------------------------+-------------------------+`
any 5 or 9 solves the puzzle
Code: Select all
`+---+---+---+|63.|241|78.||284|76.|1.3||.17|.83|624|+---+---+---+|123|8.7|.46||7.6|432|8.1||4.8|61.|237|+---+---+---+|342|178|.6.||861|..4|372||.7.|326|418|+---+---+---+`

With regard to pencil-marks, pms or preposition clues......in an empty grid, with no given clues, 9*81= 729 pms demonstrated.

In a solved puzzle there are 81 correct pms [given clues + inserted clues] [dependant on the number of clues in the puzzle]

Inital reductions are "box / line of sight" or "simple"

To solve the puzzle "round the corner" interactions have to occur, and i can understand your vision of "possible templates", indeed some solvers work this way !!

At any rate to solve a puzzle you have to get rid of 729-81 = 648 incorrect pms

With a random 16 clues, hopefully valid and minimal, you may get around 4 million grid solutions, there might not be any insertable clues and there will be many "real" pms.

In a 17, which has one solution
Code: Select all
`+---+---+---+|...|.4.|7..||.8.|..5|...||.1.|...|.2.|+---+---+---+|...|8..|..6||7..|...|...||4..|...|2..|+---+---+---+|3.2|.7.|...||...|...|...||...|..6|.18|+---+---+---+`
as you say a single clue in an otherwise empty grid can [at most] have the following effects on the pms:
in the box - 8 [for the clue position]plus 8 [for the clue in the other positions in the box] - to leave a single pm in a box - this is what an independant given clue does on its own. = 16
outside the box - 6 plus 6 "line of sight" =12

= 28pms per clue total

the first 6 clues [the maximal in this puzzle] will get rid of 28 pms each !!! = 168, you dont need a "sumation correction " yet
Code: Select all
`+---+---+---+|...|...|7..||.8.|...|...||...|...|...|+---+---+---+|...|...|..6||...|...|...||4..|...|...|+---+---+---+|..2|...|...||...|...|...||...|...|.1.|+---+---+---+`

If anything can make the approximation simpler maybe this puzzle [one of 2 known 17-puzzles with the 9clue rainbow template] will help...
Code: Select all
`+---+---+---+|1..|..6|...||...|2.7|...||9..|...|3..|+---+---+---+|.4.|.9.|...||3..|.5.|...||...|...|.6.|+---+---+---+|..7|...|2..||..6|..8|...||...|...|1.9|+---+---+---+ valid 17-puzzle+---+---+---+|1..|...|...||...|2..|...||...|...|3..|+---+---+---+|.4.|...|...||...|.5.|...||...|...|.6.|+---+---+---+|..7|...|...||...|..8|...||...|...|..9|+---+---+---+  9 clues which are independent,+----------------------------+----------------------------+----------------------------+| 1        2356789  2345689  | 3456789  346789   345679   | 2456789  245789   245678   | | 3456789  356789   345689   | 2        1346789  1345679  | 1456789  145789   145678   | | 2456789  256789   245689   | 1456789  146789   145679   | 3        1245789  1245678  | +----------------------------+----------------------------+----------------------------+| 2356789  4        1235689  | 136789   1236789  123679   | 125789   1235789  123578   | | 236789   1236789  123689   | 1346789  5        1234679  | 124789   1234789  123478   | | 235789   1235789  123589   | 134789   1234789  123479   | 1245789  6        1234578  | +----------------------------+----------------------------+----------------------------+| 2345689  1235689  7        | 134569   123469   1234569  | 124568   123458   1234568  | | 234569   123569   1234569  | 1345679  1234679  8        | 124567   123457   1234567  | | 234568   123568   1234568  | 134567   123467   1234567  | 1245678  1234578  9        | +----------------------------+----------------------------+----------------------------+`

add 8 clues to solve it.
Code: Select all
`+----------------------------+----------------------------+----------------------------+| 1        23578    23458    | 34589    348      6        | 45789    245789   24578    | | 4568     3568     3458     | 2        1348     7        | 45689    14589    14568    | | 9        25678    2458     | 1458     148      145      | 3        124578   1245678  | +----------------------------+----------------------------+----------------------------+| 25678    4        1258     | 13678    9        123      | 578      123578   123578   | | 3        126789   1289     | 14678    5        124      | 4789     124789   12478    | | 2578     125789   12589    | 13478    123478   1234     | 45789    6        1234578  | +----------------------------+----------------------------+----------------------------+| 458      13589    7        | 134569   1346     13459    | 2        3458     34568    | | 245      12359    6        | 134579   12347    8        | 457      3457     3457     | | 2458     2358     23458    | 34567    23467    2345     | 1        34578    9        | +----------------------------+----------------------------+----------------------------+`

regards
C
coloin

Posts: 1864
Joined: 05 May 2005

found the error for 125. miss counted. so it is 126.

Code: Select all
`+---+---+---+ |1..|..6|...| |...|2.7|...| |9..|...|3..| +---+---+---+ |.4.|.9.|...| |3..|.5.|...| |...|...|.6.| +---+---+---+ |..7|...|2..| |..6|..8|...| |...|...|1.9| +---+---+---+`

thanks for the puzzle. all the individual templets numbers 1- 9 solve as line of sight to a point. then i looked at each set of numbers
one at a time.

brought each of there respected rookerie down to either completion or a stop point where line of sight reduction stops leaving squares sets or combinations of squares. typically (4/6/7/8 cells)

notes.
7 is the oddity caused by overlapping squares, where 1 corner is linked.

the next number i looked solved on one of these squares on the inital number i looked at as the only single via line of sight (completed the inital grid i looked at), or solved to a second point leaving combination of 4 numbers (typically 4/6/7/8/ patterns).
repeat for each number.

basically i view these as an overlapping of 9 dimensional spaces.
each number is a dimension. limited by constraints row/column/box.

expressing the curve of possiblities (not considering variation per box as total count) but it is expressed via line of sight reduction.

0 - clue : 1 possible
1 - clue: 9 possibile
2 - clues: 36 possible.
3 - clues: 84 possible
4 - clues : 126 possible
5 - clues: 126 possible
6 clues: 84
7 clues: 36 popossiblessible
8 clues : 9 possible
9 clues : 1 possible

total = 512 ways of mapping out a single clue.

than add in a dimensional limit of no number can overlap same space for a unique solution. and thats where i found conflict of placement.

if a box/line reduction forces a zero solution in any of the 9 dimension it creates a conflict.

basic rules.

if clue 2 and clue 1 both occupy cell A,b,c,d (all in a some combination of square patterns) you have mutiple solutions. (4/6/7/8 etc) forms a none terminating error. bifucation.

if clue 2 must go in cell A(reduced via line of sight) where A is already clue 1 = 0 solutions, or vice versa. (conflict!)

if clue 2 is the only clue left in A > B,C = clue 1 > D = clue 2. or reversed. valid

heres my example from your puzzle looking only at the 9 template. @ are pm. x are dimensional numbers already set.

Code: Select all
` *-----------* |x..|@.x|.@@| |...|x.x|.@@| |9..|...|X..| |---+---+---| |.x.|.9.|...| |x@@|.x.|@@.| |.@@|...|@x.| |---+---+---| |.@x|@.@|x..| |.@x|@.x|...| |...|...|x.9| *-----------*`

basic rules line of sight - R5C2 R6C2 cannot = 9 due to r7c2,r8c2 pm

R1c4 = 9 dimensional constraint(all the x's limit 3 spaces) + line of sight(r3c1,+r4c5) > R7c6 = 9 > R1c8,r1c9 cannot = 9 (via line of sight of R1c4) > R7c2 cannot = 9. therfor R8c2= 9(line of sight + dimensional)

which solves to this point. leaving 2 linked squares (7 cells)
Code: Select all
` *-----------* |x..|9.x|...| |...|x.x|.@@| |9..|...|X..| |---+---+---| |.x.|.9.|...| |x.@|.x.|@@.| |..@|...|@x.| |---+---+---| |..x|.9|x..| |.9x|..x|...| |...|...|x.9| *-----------*`

at that point i would search each other grid for a single that would solve one of those 7 cells which would force the above squares to compeltion.
the order i pic is the 6 grid followed by 7 > solves the 9 grid. then repeat and the puzzle solves.
Code: Select all
` *-----------* |x..|x.6|...| |@@.|x.x|@.@| |x@.|...|x.@| |---+---+---| |@x.|@x.|...| |x@.|@x.|...| |...|...|.6.| |---+---+---| |..x|@@x|x.@| |.x6|..x|...| |...|@@.|x.x| *-----------* `

r7c9 = 6 (line + dimensioal) > r2c7 = 6 (line + dimensional) >
r3c2 = 6 (line + dimensional) > r4c1 = 6 (line + dimensional) >
R5c4 = 6 (line +dimensional) > r8c5 = 6 (line + dimnsional)

Code: Select all
` = complets all off the 6 grid.*-----------* |x..|x.6|...| |...|x.x|6..| |x6.|...|x..| |---+---+---| |6x.|.x.|...| |x..|6x.|...| |...|...|.6.| |---+---+---| |..x|..x|x.6| |.x6|..x|...| |...|.6.|x.x|`
note the location r2c7 = 6 > forces r2c8 = 9 further complets parital 9 grid.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

Im not sure what you are achieving here.

We know this puzzle with 17 clues has a unique solution, this puzzle has 9 of the rainbow template - plus 8 clues.
We know that it wont have a unique solution without all the clues , it will only partially complete with less clues.

Perhaps given the nine initial clues you could look at the pm reductions [or other analysis] achieved with each of the ways you can add the other 8 clues....

ways to add 1 more clue = 8
ways to add 2 more clues = 28
ways to add 3 more clues = 56
ways to add 4 more clues = 70
ways to add 5 more clues = 70.....

Approaching 9 plus 8 clues will introduce non line of sight reductions - these ultimately solve the puzzle, but become undefinable.

C
coloin

Posts: 1864
Joined: 05 May 2005

Approaching 9 plus 8 clues will introduce non line of sight reductions - these ultimately solve the puzzle, but become undefinable

i belive they are definable.
im suggesting this.

the grids 1-9 expressed as a templet of 9 digits are planes or dimenions
each plane has box/line of sight rules attached to them. (first simplistic rule of coding)

then u have a new rule added for planer interaction.

each plane displaces or restricts the placement of each digit by eliminating space. either via a real placed number or super imposed pm of remaining digits from other planes. so u can have 1+1 plane as i showed in my previos post ,interaction, 3 planes all locked together. 4 & 5 planes locked, perhaps 6/7 but rare.

when a placed or pm digit restricts the placement of a digit on another plane down to a location of each box to having no vaild placement. then a invaild puzzle exists. (zero solutions). (im perposing they can be defined befor they happen)

when a set of digits are not restricted beyond a specific point ( ie completing one corner on a square) then mutiple solutions remain. (if this happens as in the case of 16 clue puzzle - than mathmatically all subsets of known "conflicts" cannot restrict enough penicl marks to produce a single solution only mutiple.

so the question is what are all the known conflicts of placements that would induce when completed a invalid solution (thus a vaild puzzle avoids them)

each combination of planer interactions cann't have a zero placement issue. where a set of given pms+clues invalidates or produces a zero placement cell.

for example

Code: Select all
`+---+---+---+ |12.|...|...| |...|...|...| |...|...|...| +---+---+---+ |21.|...|...| |...|...|...| |...|...|...| +---+---+---+ |..?|...|...| |...|..2|..1| |...|..1|..2| +---+---+---+ `
? = both the 2 and the 1 in the same singular cell.
where if any of the 2/1 sets are force or placed and compelted in said pattern a zero solution happens. (example of bi-planer interaction) how would u prevent said pattern from happening? a conflict check befor hand would work

im writing out forced restrictions patterns, where cells cann't be a specific number or a zero solution happens.
(which is where im mostly looking as the definintion of these restrictions prevent the zero placement from happening to begin with)

fore example.
Code: Select all
` *-----------* |12.|...|...| |...|12.|...| |...|...|12.| |---+---+---| |21.|...|...| |...|21.|...| |...|...|21.| |---+---+---| |..1|..X1|..C| |..A|..1|..D| |..B|..x2|..1| *-----------* `
where if x1+a is force to = 2 box/quad 9(c/d) = 0

ie A+c = 2 : vaild (solves x2) or
(issomorphic codvariations : a+x2(solves c), or C+x2 (solves a)

b +x1 = 2 : vaild (solves C) or issormorphic code variations : c+b(solves x1), or x1+C (solves b)

x1 + D = 2 vaild (solves B ) or (issomorphic code variation : x1+ b = 2(solves d) or d+b(solves x1)

avoiding having x1+a = 2 at the same time, across the 9 dimension interaction is what im looking to do. (this prevents the zero solution)

menaing that x1+a cannot be solved as 2 at once. avoiding that subset is the essence of P.

thus a restriction is created when the gives(mapped above) are set a & x1 cannot = 2.
P applies to all cells that direectly line of site affect the placemetn of x1 and a. so that if any portion of the chain of placements reduces half of the affected cells and forces A or x1 to equal to 2 the opposit side is equal to not 2 to prevent the zero,

so basically u have to check the interaction of all planes so that none of them restict the placemetn of A&X1 as only a 2
thus P exists and can be defined.

not sure if that makes any sence. but thats my idea.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

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