22.04.2020, 13:28
I would like to generalize this technique for all latin squares (regions doesn't play a role if you formulate a bit differently) and choice of x rows and y columns.
The technique can be seen as dissections of the grid in 4 sets of cells, and it is interesting to see how the latin square rules act.
Make a choice of x rows and y columns. Let's take your example: rows 1289 and columns 1289. Let's say the Red (R ) set of cells for intersecting cells of these rows and columns.
Then you can define the Green (G) set as intersecting cells of complementary rows and complementary columns (rows and colmuns 34567 in our example).
Then the set R and G are the same up to a number of whole set of digits 1-9 (Let's call this set S).
But we can also define the Yellow (Y) set: intersecting cells of chosen rows and complementary columns (in our example rows 1289 and columns 34567),
And Blue (B) set as intersecting cells of complementary rows with chosen columns (rows 34567 columns 1289).
Then we have the same conclusion: Y=B mod(S) Yellow set of cells contain same digits as Blue set, modulo whole set of digits 1-9.
It is very easy to understand: R+Y give complete rows R+Y=nS and G+Y give complete columns G+Y=mS, then R=G+(n-m)S or R=G modulo S. Same for Y and B: Y=B modulo S.
The number of combination of different dissections in the grid is impressively high (15'876 combinations if you chose 4 rows and 4 columns, but you can chose any number of rows and any number of columns). This is why probably this technique is not practicable for human solvers (You can't check such big number of dissections).
You can make fun dissections of the grid:
Like first example: Red set of digits is the same as Green set, and Yellow set contains Blue set + complete set of digits 1-9.
Another funny observation is that you can build X-wing and fish techniques (swordfish, jellyfish) with this technique:
If a digit is not in the red set, it is not in the green set (X-wing), in other words, it appears twice in yellow set.
Same for swordfish here.
Fred
The technique can be seen as dissections of the grid in 4 sets of cells, and it is interesting to see how the latin square rules act.
Make a choice of x rows and y columns. Let's take your example: rows 1289 and columns 1289. Let's say the Red (R ) set of cells for intersecting cells of these rows and columns.
Then you can define the Green (G) set as intersecting cells of complementary rows and complementary columns (rows and colmuns 34567 in our example).
Then the set R and G are the same up to a number of whole set of digits 1-9 (Let's call this set S).
But we can also define the Yellow (Y) set: intersecting cells of chosen rows and complementary columns (in our example rows 1289 and columns 34567),
And Blue (B) set as intersecting cells of complementary rows with chosen columns (rows 34567 columns 1289).
Then we have the same conclusion: Y=B mod(S) Yellow set of cells contain same digits as Blue set, modulo whole set of digits 1-9.
It is very easy to understand: R+Y give complete rows R+Y=nS and G+Y give complete columns G+Y=mS, then R=G+(n-m)S or R=G modulo S. Same for Y and B: Y=B modulo S.
The number of combination of different dissections in the grid is impressively high (15'876 combinations if you chose 4 rows and 4 columns, but you can chose any number of rows and any number of columns). This is why probably this technique is not practicable for human solvers (You can't check such big number of dissections).
You can make fun dissections of the grid:
Like first example: Red set of digits is the same as Green set, and Yellow set contains Blue set + complete set of digits 1-9.
Another funny observation is that you can build X-wing and fish techniques (swordfish, jellyfish) with this technique:
If a digit is not in the red set, it is not in the green set (X-wing), in other words, it appears twice in yellow set.
Same for swordfish here.
Fred