by **dukuso** » Thu Jul 07, 2005 7:33 am

Let S be the collection of the 84 subsets of {1,..,9} of size 3.

Let T be the collection of the 4741632000 9-tupels (S1,..,S9)

with members from S such that

S1+S2+S3=S4+S5+S6=S7+S8+S9={1,2,3,4,5,6,7,8,9} ,

where "+" is set-union.

Let 2 members from T be equivalent if one can be obtained

from the other by one of the known obvious 6^4*9! operations.

Then we get the collection E of the known 44 minimal elements from T,

one from each equivalence class, called "the gang of the 44".

For M in T let C(M) be its equivalent representative in E.

For M in E let Z(M) be the size of its equivalence class.

For M in E let N(M) be the number of partial 3*9 sudoku grids

compatible with M with respect to the 6^9 operations

of permuting inside a column.

The 2*44 values of N(M),Z(M) are constant within one class and

can be calculated quickly.

For each M=(M1,..,M9) in T let U(M) be the set of the 175616

3-tupels (M,A,B) in T^3 , A=(A1,..,A9),B=(B1,..,B9)

such that Mi+Ai+Bi={1,2,3,4,5,6,7,8,9} for each 1<=i<=9.

----------------------------------------------------------

then the total number of sudokus is the sum over all M in E,

over all (M,A,B) in U(M)

of Z(M)*N(M)*N(C(A))*N(C(B)).

----------------------------------------------------------

This indeed adds up to 6670903752021072936960.

There are 7727104 summands in total, but since many of the

N(x) and Z(x) coincide, only 1398 summands are different.

You can easily save a factor of 2 by discarding triples

in U(M) with A>B.

Maybe another factor of 3 can be saved by requiring M<A<B,

but that makes things more complicated.

How can we calculate the total number of sudoku-classes

as defined by RedEd but discarding transposition

with this method ?

sum over all M in E,

over all (M,A,B) in U(M)

of Z(M)*Q(M)*Q(C(A))*Q(C(B))

gives 19859770556

E = "gang" of the 44

=====================

(1) M, element of E, minimal in its class ("ganger" or "gangster" ?)

(2) C(M)

(3) N(M) , number of compatible 3*9 sudokus (bands)

(4) Q(M) number of classes of compatible bands (from the gang of 416)

(5) Z(M) , number of 9-tupels from T in the class

--------------------------------------------------

123456789123456789123456789,1,1728,2,60480

123456789123456789123457689,2,576,3,4898880

123456789123456789124357689,3,192,3,29393280

123456789123456789124378569,4,192,2,9797760

123456789123456789147258369,5,96,2,6531840

123456789123457689123458679,6,576,4,6531840

123456789123457689123468579,7,864,7,6531840

123456789123457689124356789,8,192,6,58786560

123456789123457689124358679,9,192,8,117573120

123456789123457689124367589,10,192,6,58786560

123456789123457689124368579,11,288,24,235146240

123456789123457689124389567,12,192,6,58786560

123456789123457689126345789,13,192,3,29393280

123456789123457689126347589,14,192,8,117573120

123456789123457689126348579,15,288,16,117573120

123456789123457689145267389,16,96,1,29393280

123456789123457689145268379,17,144,8,117573120

123456789123457689146258379,18,168,14,235146240

123456789123457689148259367,19,144,3,58786560

123456789124357689125348679,20,192,5,39191040

123456789124357689125367489,21,144,4,58786560

123456789124357689125368479,22,168,9,117573120

123456789124357689125378469,23,264,13,117573120

123456789124357689126358479,24,144,8,117573120

123456789124357689126378459,25,168,14,235146240

123456789124357689126389457,26,144,4,58786560

123456789124357689128345679,27,192,10,117573120

123456789124357689128356479,28,168,9,117573120

123456789124357689128359467,29,120,4,78382080

123456789124357689134258679,30,192,5,39191040

123456789124357689134268579,31,288,28,235146240

123456789124357689135268479,32,228,22,235146240

123456789124357689135278469,33,276,26,235146240

123456789124357689136258479,34,168,9,117573120

123456789124357689136278459,35,180,30,470292480

123456789124357689137268459,36,216,20,235146240

123456789124357689138259467,37,156,26,470292480

123456789124357689138269457,38,228,11,78382080

123456789124357689158267349,39,120,6,117573120

123456789124378569129356478,40,96,2,3265920

123456789124378569135279468,41,192,8,78382080

123456789124378569137245689,42,516,9,26127360

123456789124378569157268349,43,168,6,39191040

123456789147258369159267348,44,120,2,4354560

-------------------------------------------------

sum:11352,416,4741632000

60713 out of the 85184=44^3 tupels of 3 members

from the list can be joined to form a valid 9*9 sudoku.

-Guenter Stertenbrink, sterten(-at-)aol.com