## Mathematical Recreations

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This topic concerns sudoku puzzles with arrangements of six or nine 3×3 boxes, such as those posted as You can find all of the puzzle configurations in the sudoku archive and some other variations, too.

#### Sudoku Variants

These puzzles are composed of 3×3 boxes, just like the classic sudoku. But it is not necessary to arrange the 3×3 boxes in a 3×3 grid. In fact, other arrangements make truly excellent puzzles, and each arrangement has its own flavor.

Consider the puzzle below, composed of six 3×3 boxes:

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

The puzzle has 3×3 boxes with nine cells each. There are twelve strips with nine cells each, and each strip passes through three 3×3 boxes. These strips are just like the rows and columns of the classic sudoku, but a whole new world is opened up when rows can intersect rows and columns can intersect columns.

This simple change creates more types of interactions. For example, if you try to place a three in column six, you'll notice that the three in the fourth row eliminates two cells—the one in the fourth row, and the one in the second column:

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

This is only a little change from classic sudoku, but the same type of configuration works at the next level of inference, too. Consider the placement of nines in row one. There are two possible locations for the nine—either in column four or in column seven:

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

But cell (4,4), i.e., the intersection of the fourth row and the fourth column, neighbors both of these cells. As a result, you cannot place a nine in cell (4,4). You can understand this two ways. If you place a nine in cell (4,4), then there will be no place left for a nine in row one. Alternatively, regardless of where you place a nine in row one, you cannot place a nine in cell (4,4).

Enjoy the puzzle above. It doesn't require the type of inference that I just described, but other puzzles of this type can be more difficult. I'll be posting more of these puzzles, and other variants, on the Sudoku of the Day page. Some of the other puzzle types have additional interactions between the rows, columns and boxes. It's also possible to build puzzles from 2×2 or 4×4 boxes. For example, Cube Sudoku is the only puzzle of this type with six 4×4 boxes.

It's interesting to consider what sudoku variants can be made by joining 3×3 boxes. If you keep the strips (i.e., rows and columns) in groups of three, where each group of three strips passes through the same three 3×3 boxes, then there are a limited number of puzzle configurations that you can make from a fixed number of 3×3 boxes. Classic sodoku with nine 3×3 boxes is one, and the wedge configuration above with six 3×3 boxes is another. What is the smallest puzzle that you can make in this way? How many sudoku configurations can you make with no more than nine 3×3 boxes? (To be clear, I'm talking about puzzles where each cell is in exactly three groups: a 3×3 box, a strip passing through the box vertically, and another strip passing through the box horizontally. Each strip passes through three 3×3 boxes, and there are two other parallel strips that pass through the same three 3×3 boxes.)

View the Solution

I've made sudoku puzzles with all of the possible arrangements of six or nine 3×3 boxes, and posted them as Sudoku of the Day. (Well, I haven't featured a classic sudoku puzzle as Sudoku of the Day, but we all know what that looks like.) You can find all of the puzzle configurations in the sudoku archive and some other variations, too.