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 dont understand your use of the word "quad"
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/
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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.
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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
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*-----------*
|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.
