thats y i said it was the easiest to link a basic example that doesn't show constraits (im thinking that math example simple looks at total number of possiblities)
9!x8!x7!x6!x5!X4!x3!x2!x1! (none restraint application) with derivitives shows 16.xxx rounded = 17
and isn't this an incorect interpretation of the number of puzzles?
1.834933472 x10^21
where there is
http://www.afjarvis.staff.shef.ac.uk/sudoku/bertram.htmli cannot valid that it wasn't a joke post either he simply asked for a link on it for an example. (i used a link for an article i found on the net rather then posting way too many pages of math by hand)
some of the solid numbers he / she uses in the artical i can't figure out where they got them from either..
but any way a
restrained application would have to be formed like this.
first
find the equation to show restraints
in reality its (where n is the number of placed variables to the order of 9)
where n = (1:9) i do it by placing individual numbers 1 at a time.
where n is placed 9 times.
to show the number of remaining positions where n can only be placed(didn't consider defining each place just represented it to a specific power)
lets say you place all nine of the ones on the board.
you'd mathmatically reduce expoentially spaces, as follows.
(placing numbers as singles)
N1^81
N2^(81 - 21) (21 = number of placements the 1st # removed)
N3^(81 - 36)
N4^(81 - 45)
N5^(81 - 58)
N6^(81 - 67)
N7^(81 - 72)
N8^(81 - 77)
N9^(81 - 80)
next start placing the "2"s 9 times. (heres where a demstration of minimal clues needed arives)
N10^(81 - 9)
N11^(81 - 28)
N12^(81 - 41)
N13^(81 - 48)
N14^(81 - 61)
n15^(81 - 70)
N16^(81 - 75)
right up to here you can freely place 16 numbers any where and not generate a single valid sudokus (as placement conflicts are not present)
but once you place the 17th number the grid's limits placements increasingly bye 1^n positions with each addinal placement
don't belive me here
try this easy example
place all the 1's as follows. start from the top place a one in box 1.
my math is shown. then in quad 2. repeat till you place all the 1's
now go and do the same for the 2's... keep track of my mathmatical representation.
continue untill you reach the 16's number placed. (checks its math) now move to the last postion indicated with an x. (x postion is actually invalid as (2's+1's in last grid renders x as an invalid postion)
- Code: Select all
*-----------*
|12.|...|...|
|... |12.|...|
|... |... |12.|
|---+---+---|
|21.|... |...|
|... |21.|...|
|... |... |21.|
|---+---+---|
|..1|..x|...|
|..2|..1|...|
|...|...|..1|
*-----------*
N17^(81 - 78) is actually a false representation of valid positions. overlapping 2+1's limit this number bye +1 additional square.
for every number beyond 16 placed the math changes respecfully.
since this is the first positon where overlaps can render additional clues
pointless (17 is the minmual amount of clues needed)
after 16 clues are placed addition clues placement changes in respect to the invalid(position constraints)
a more actuall equation would be
N^(81 - invalid positions - (spacial invalid)
shown as follows for the 17th position.
n17^(81- 78-(1) where 1 = overlap)
n18^(81- 80)
