Hi,

eleven!

eleven wrote:Serg wrote:Please, clarify - did you calculate sample patterns batch only or all possible patterns (I suspect the first is true)?

This list should include all possible valid patterns, if i made no mistake. It was not hard to calculate it, so it should not be a big problem for others to confirm that or find missing ones.

I tried to calculate 18-clue symmetric patterns' number too.

Though it is possible to calculate number of

essentially different valid 18-clue symmetric patterns accurately, but it needs some time, so I present my calculation of estimated low bound of 18-clue symmetric patterns only. All calculations were done manually, so they can be cross-checked easily.

I consider such possible isomorphic transformations:

1. Bands permutations (6 ways).

2. Rows permutations within the upper band (6 ways).

3. Rows permutations within the middle band (6 ways).

4. Rows permutations within the lower band (6 ways).

5. Correlated (mirrored) columns permutations within left

and right stacks (6 ways).

So, alone pattern has at most 6^5 = 7776 automorphisms.

To calculate the number of possible patterns we must consider 5 cases: central column (c5) may contain 0, 2, 4, 6 or 8 clues. We should calculate number of patterns for all cases and then to sum that numbers to obtain total number of possible patterns. Number of possible patterns in the left half of sudoku grid (rows r1-r9, columns c1-c4) is described by formula n!/(k!*(n-k)!), where n = 36 - number of cells in the left half of sudoku grid, k - number of clues in the left half of sudoku grid. This number must be multiplied by number of placing clues in column c5 - m!/(p!*(m-p)!), where m = 9 - number of cells in the c5 column, p - number of clues in the c5 column.

1. Column c5 contains 0 clues. k = 9, p = 0; number of patterns = 94143280.

2. Column c5 contains 2 clues. k = 8, p = 2; number of patterns = 30260340*36 = 1089372240.

3. Column c5 contains 4 clues. k = 7, p = 4; number of patterns = 8347680*126 = 1051807680.

4. Column c5 contains 6 clues. k = 6, p = 6; number of patterns = 1947792*84 = 163614528.

5. Column c5 contains 8 clues. k = 5, p = 8; number of patterns = 376992*9 = 3392928.

Total numbers of 18-clue symmetric patterns: 2,402,330,656 (accurate calculations).

Low bound of

essentially different 18-clue symmetric patterns (as if each pattern would have 7776 automorphisms) - 2402330656/7776 = 308941 (approx.)

This estimation correlates with your number. Maybe I'll calculate accurate number of 18-clue symmetric patterns in some time.

eleven wrote:Though i am interested to do more, it is not possible for me to invest more than a few hours per week for that. So from my side there will not come much input for a new thread.

In the moment i am interested in a simple idea to fasten the exhaustive search. I hope that i can test this week, if its of any worth.

So am I

. I agree, it's not the time to open separate thread devoted to 18-clue symmetric patterns exhaustive search.

Serg

[Edited: I corrected an error in my calculations.]

[Edited2: I corrected an error in number of automorphisms calculations.]

[Edited3: I corrected an error in arithmetic calculation. Total numbers of 18-clue symmetric patterns is still 2,402,330,656 (as

eleven wrote).]