a setokugrid is a bijective map s=(s_1,s_2,s_3,s_4):I3^4-->I3^4 such that
s_i(-,b,c,d)=s_i(a,-,c,d)=s_i(a,b,-,d)=s_(a,b,c,-)=I3
for all a,b,c,d in I3 , i in I4 ^.
write the 4 digits s(a,b,c,d) in a cell of a 9*9-grid at position (a*3-3+b,c*3-3+d)
or arrange the SET-cards in a 9*9-array, such that the 3 positions in
the 6 sets 123,456,789,147,258,369 in each row and each column contain
all the 3 expressions of all the 4 properties.
there are 75168 setokugrids from 29 symmetry-classes
make a puzzle by removing cards ...
but presumably these don't give so nice puzzles for handsolvers
since you have to check lots of the remaining cards
at each move
this is again an exact cover problem, so the exact-cover-computer-solvers
can be easily adapted
maybe I can create some puzzles later
1st of the 75168 solution grids:
- Code: Select all
2
+--------+--------+--------+
|11 22 33|22 33 11|33 11 22|
|11 22 33|23 31 12|32 13 21|
| | | |
|22 33 11|33 11 22|11 22 33|
|32 13 21|11 22 33|23 31 12|
| | | |
|33 11 22|11 22 33|22 33 11|
|23 31 12|32 13 21|11 22 33|
+--------+--------+--------+
|23 31 12|31 12 23|12 23 31|
|22 33 11|31 12 23|13 21 32|
| | | |
|31 12 23|12 23 31|23 31 12|
|13 21 32|22 33 11|31 12 23|
| | | |
|12 23 31|23 31 12|31 12 23|
|31 12 23|13 21 32|22 33 11|
+--------+--------+--------+
|32 13 21|13 21 32|21 32 13|
|33 11 22|12 23 31|21 32 13|
| | | |
|13 21 32|21 32 13|32 13 21|
|21 32 13|33 11 22|12 23 31|
| | | |
|21 32 13|32 13 21|13 21 32|
|12 23 31|21 32 13|33 11 22|
+--------+--------+--------+
I think the exact cover matrix has 6561 rows and 1458 columns, much
bigger than normal sudoku with 729 rows and 324 columns.
each placement of one of the 81 "cards" into one of the 81 positions is a row.
one 1 goes into 81 cells so each cell gets exactly one symbol
one 1 goes into 81 symbols=cards so each card is use exactly once
one 1 goes into one of 3 columns for each of 27 minirows for each of 4 3ary digits ("properties")
minicolumns
distance3subrow
ditance3subcolumn