so it can be found with the forum's search function.
Maybe we can gather all the important posts here
in this thread for referrence and easy linking.
If you want to reply or discuss, please reply to the original thread:
http://forum.enjoysudoku.com/viewtopic.php?t=44&postdays=0&postorder=asc&start=454
--------------------------------------------------------
The number of S-classes and T-classes as well as (once again)
the number of grids for 3x3-sudoku [81 cells] is now confirmed.
Let A be the subgroup of S_9xS_81 of transforms acting on 3x3-sudoku grids,
generated by symbol permutations in S_9x{Id}, band permutations,
stack permutations, permutations of rows within a given band and of columns
within a given stack. [we omit transposition here]
For any grid G, let K(G) be the transforms in A leaving G fixed.
[the "isotropy-group" of G]
And let T[G] be the equivalence class of G modulo A.
[the "orbit" of G by the action of A = the T-class of G]
Then #T[G] = #A/#K(G) = 9!*6^8/#K(G)
I have written a program that runs through all T-classes,
and for each class T[G] (where G is a grid) finds the size of K(G),
and whether G is transpose-invariant (i.e. T[G]=T[transpose(G)]).
The result of the counting:
- Code: Select all
i | N | X | Y
---|----|-------------|-----------
| |Classes T[G] with #K(G)=N
| |not transpose| transpose
| | invariant | invariant
---|----+-------------+-----------
1 | 1 | 10944340774 | 23201
2 | 2 | 1050496 | 637
2 | 3 | 14672 | 36
4 | 4 | 4378 | 29
5 | 6 | 2442 | 6
6 | 9 | 84 | 1
7 | 12 | 172 | 0
8 | 18 | 168 | 4
9 | 27 | 4 | 1
10 | 36 | 22 | 2
11 | 54 | 20 | 1
12 |108 | 4 | 0
13 |162 | 2 | 0
14 |324 | 0 | 1
---|----|-------------|-------
|sum:| 10945413238 | 23919
Let the entries in above table in row i be denoted by (N_i,X_i,Y_i) then
The number of T-classes = Sum (X_i + Y_i) = 10945437157
The number of S-classes = Sum (X_i/2 + Y_i) = 5472730538
The number of S-classes that split into two T-classes = Sum X_i/2 =
5472706619
The number of S-classes with only one T-class = Sum Y_i = 23919
The number of grids = 9! * 6^8 * Sum (X_i+Y_i)/N_i = 6670903752021072936960
The number of transpose-invariant grids = #A * Sum Y_i/N_i =
14347728127918080
