7 (7) 2 8 4 5 3 6 9 3 8 5 7 4 1 2 9 3 5 2 2 4 1 5 6 2 4

#### Two-Block Interactions

Sometimes you can't place a value in a particular cell, but you know it has to go in one of two or three possible locations. Let's try to place the value seven in column 6 of this puzzle. Each of the yellow cells is in the same row as a seven or it is filled with a value other than seven. So we know that one of the two green cells has to hold the seven in column 6. Since both of these cells are in box 8, we know that none of the red cells in box 8 can hold a seven. Alternatively, we can see that any attempt to place a seven in any of the red cells would mean that there would be no place left to put a seven in column 6.

As a result, the seven on column 4 must go in row 2, since all other possibilities are eliminated.

 7 2 8 4 (7) 5 3 6 9 3 8 5 7 4 1 2 9 3 5 2 2 4 1 5 6 2 4

Now look the puzzle again. Let's try to place a seven in box 5. There are two possible locations. Since both of these locations are in column 5, we cannot place a seven in any of the red cells in column 5.

As a result, the seven in row 3 must go in column 2, because all other locations have been eliminated. You can also determine that the only location for a seven in box 2 is in cell (2,4), which is the same conclusion that we made above. And cell (3,5) must be an eight, because all of the other nine values have been eliminated for that cell.

#### Three-Block Interactions

In classical 9×9 sudoku with 3×3 boxes, you only have block interactions when a value is restricted to the intersection between a 3×3 box and either a row or a column. This is a two-block interaction, because it involves the intersection between two blocks. But this is not the most general type of block interaction. To understand the more general case, let's look at block interactions a little differently.

Look at the two examples above, and consider any red cell. If that red cell were given a value of seven, then it wouldn't leave any valid cells to hold a seven in the column or box. We say that all valid cells for a seven in the column or box are neighbors of the red cell.

In classical sudoku, there are 27 blocks (9 rows, 9 columns and 9 boxes). Each cell is a member of three blocks (one row, one column and one box). Each cell has 20 neighbors: six in the same row (but not in the same box), six in the same column (but not in the same box), four in the same box (but not in the same row or column), two in the same row and the same box, and two in the same column and the same box. Now consider one of the other 24 blocks that the cell is not a member of, and consider the neighbors of the cell that are in that block. There are 4 cases:

1. There are two rows and two columns that pass through the cell's box. Each of those rows and columns has three of the cell's neighbors in the intersection between the row or column and the cell's box.
2. There are six rows and six columns that do not pass through the cell's box. Each of those rows and columns has one of the cell's neighbors in the intersection between the row and the cell's column or between the column and the cell's row.
3. There are four boxes that intersect the cell's row or the cell's column. Each of those boxes has three of the cell's neighbors in the intersection between the box and the cell's row or column.
4. There are four boxes that do not contain any of the cell's neighbors

Now, suppose we try to place the value four in one of these 24 blocks. In case 1, we are trying to place the four in a row or column that passes through our cell's box. Let's suppose it's a row. If we know that the only possible locations for the four in that row are also inside our cell's box, then our cell cannot hold a four. If our cell held a four, then it would be impossible for any cell in the row to hold a four. This is the first example above.

In case 2, we are trying to place the four in a row or column that does not pass through our cell's box. Again, let's suppose it's a row. The only cell in this row that neighbors our cell is the one in the same column as our cell. If we know that the only possible location for the four in that row is in our cell's column, then our cell cannot hold a four. Of course, if there is only one possible location for the four in that row, then that cell must hold the four by simple elimination. This is not normally considered a block interaction.

In case 3, we are trying to place the four in a box that intersects our cell's row or column. Let's suppose it intersects our cell's row. If we know that the only possible locations for the four in that box are also inside our cell's row, then our cell cannot hold a four. This is the second example above.

In case 4, we are trying to place the four in a box that does not intersect our cell's row or column. There are no possible block interactions between our cell and this box.

A block interaction can take place when there are one or more cells outside the block that neighbor every valid location for a value inside the block. In case 1, our cell is outside the row, but it neighbors every valid location for a four in the row. In case 3, our cell is outside the box, but it neighbors every valid location for a four in the box. With this understanding of block interactions, we can talk about three-block interactions

 5 6 1 5 6 3 6 4 5 4 5 6 (3) 4 4 1 5 7 5 5 2

In this 7×7 jigsaw sudoku, the only valid locations for a four in the middle box are shown in green. The other locations in the box already hold a value, or they are in the same row as the four. Note that these three green cells are not in a single column or in a single row. Nonetheless, the red cell does neighbor each of these three cells. If the red cell held a four, then there would be no valid location for a four inside the middle box; therefore, we know that the red cell cannot hold a four.

In this case, the red cell also neighbors every other value except three; therefore, the red cell must hold a three.

I refer to this as a three-block interaction, because all of the valid locations in the box are in either the row or the column of the red cell. The row, column and box are the three blocks that interact. You can also have the situation where a cell neighbors the only valid locations in a row, but those locations are not all in the same box. This would occur in this puzzle if the only valid locations in the fourth row were the red and two green boxes. Then the green box in the second row would neighbor each of those three cells because the two green cells are in the same box and the red cell is in the same column.

Cube sudoku has very rich three-block interactions. Every row intersects every column in two different locations on opposite faces of the cube; in addition, the rings form a third set of strip blocks, and the rings intersect both rows and columns. A sudoku variant will support three-block interactions if it is possible to have three blocks (such as rows, columns and boxes) that intersect pairwise (A intersects B; B intersects C; and C intersects A), but none of the three blocks completely contains all of the intersections. In classical sudoku, if a row and a column both pass through the same block, then the row and the column intersect inside that block. As a result, only two-block interactions are possible in classical sudoku. Puzzles with three-block interactions include the Sudoku of the Day puzzles on this site, Cube Sudoku, Jigsaw Sudoku (AKA Geometry Sudoku), and Sudoku X (AKA diagonal sudoku, AKA kokonotsu).

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Thanks,
Steve