Many times in Sudoku Puzzles, you encounter what I call "Sudoku Walls".
These are rows or columns of three numbers within a single box. For example:
||===|===|===||===|===|===||===|===|===||
|| A | B | C || y | y | y || x | x | x ||
||---|---|---||---|---|---||---|---|---||
|| y | y | y || | X | || | | ||
||---|---|---||---|---|---||---|---|---||
|| x | x | x || | | || | Y | ||
||===|===|===||===|===|===||===|===|===||
This is a single Box-Row where A, B, C are known, and all are in the same row of a single box.
Then assume that X and/or Y are other numbers in different rows of the Box-Row. The locations
of the other two X's or Y's are forced into the positions marked by corresponding "x" and "y".
So, if ABC were 715, and X was 4 in B2, the other 4's would have to be in R3 of B1 and R1 or B3.
Similarly, if Y was 6 in B3, the other 6's would have to be in R2 of B1 and R1 of B2.
Of course, you could flip this Box-Row up onto either end, and you'd have a Box-Column.
Here is a sample puzzle with a few walls, and where others develop as you begin to solve it.
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|| 7 | | 5 || | | || | | 4 ||
||---|---|---||---|---|---||---|---|---||
|| | | || | | || | | 3 ||
||---|---|---||---|---|---||---|---|---||
|| | | 9 || 4 | | 8 || 5 | | ||
||===|===|===||===|===|===||===|===|===||
|| | 2 | || 7 | | || | 1 | ||
||---|---|---||---|---|---||---|---|---||
|| | | || 6 | | || | | ||
||---|---|---||---|---|---||---|---|---||
|| | 1 | || | | 4 || | | ||
||===|===|===||===|===|===||===|===|===||
|| | | 4 || 5 | 7 | 3 || 2 | | ||
||---|---|---||---|---|---||---|---|---||
|| 5 | | || | | || 4 | | ||
||---|---|---||---|---|---||---|---|---||
|| 1 | | || | | || 9 | | 8 ||
||===|===|===||===|===|===||===|===|===||
There are two walls already given: the top row in B8, and the left column in B9.
You still use standard isolation methods to find things, like the "1" at (R2,C3), which creates
another wall in the right column of B1. The wall in B8 helps your find a "1" in B9, which
has to be at (R7,C9). Of course, a What's Left List of 1,6,8,9 for R7 also finds that same "1",
and then the "6" at (R7,C8). From the two newly found 1's, you find another at (R1,C7), etc.
Notice how many pairs of numbers there are within a box in the same row or column.
The 7,5 in R1, the 4,8 in R3, the 9,8 in R9, the 5,1 in C1, the 2,1 in C2, the 5,9 in C3,
the 7,6 in C4, and the 4,3 in C9. Find just one more number to complete the trio (ABC),
and you'll probably create another pair which can turn into a trio. Trios can be very helpful.
Here's another sample, but this one has two row-walls in a single box-row, which has an
interesting consequence.
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|| 2 | 4 | || 9 | | || | | ||
||---|---|---||---|---|---||---|---|---||
|| | 1 | 8 || 3 | | || 5 | | ||
||---|---|---||---|---|---||---|---|---||
|| | | || | | 8 || | | ||
||===|===|===||===|===|===||===|===|===||
|| | | || | | || 6 | 5 | 2 || L{4,7,8},M{1,3,9}
||---|---|---||---|---|---||---|---|---||
|| | | || | | || | | || L{2,5,6},M{4,7,8},R{1,3,9}
||---|---|---||---|---|---||---|---|---||
|| 1 | 9 | 3 || | | || | | || M{2,5,6},R{4,7,8}
||===|===|===||===|===|===||===|===|===||
|| | | || 2 | | || | | ||
||---|---|---||---|---|---||---|---|---||
|| | | 5 || | | 6 || 8 | 7 | ||
||---|---|---||---|---|---||---|---|---||
|| | | || | | 7 || | 1 | 4 ||
||===|===|===||===|===|===||===|===|===||
The two walls in rows 4 and 6 contain completely different numbers (no duplicates). Therefore, the
1,9,3 on the left must be repeated, in any order, under the 6,5,2 on the right of row 5, and to the
left of 6,5,2 in the middle of row 4. Likewise, the 6,5,2 in row 4 on the right must be repeated, in
any order, above the 1,9,2 on the left of row 5, and to the right of 1,9,2 in the middle of row 6.
That leaves 4,7,8 in any order, in three specific places: row 4 in the left box, row 5 in the middle box,
and row 6 in the right box. All of these triples are "phantom finds", meaning you know where they go,
but not their order in their three cells. If you create "what's left lists" at the end of these rows, group
them in brackets marked "L", "M", and "R", and in that order. (see the picture above)
To understand phantoms better, visit "Sudoku Phantoms".