In the following we first describe a connection of -nets and Sudoku and then generalize the Sudoku in analogy to higher order -nets in base 3. A new type of Sudoku puzzle is presented at the end.

A -net in base 3 is a set of 9 points in the square which have the following property:

Each rectangle

for nonnegative integers , , , contains exactly one point of the point set.

Such point sets can be generated using Sudoku. Consider the following example of a Sudoku: The square can be partitioned into 9 rows, each containing the numbers 1 to 9, it can be partitioned such that each column contains the numbers 1 to 9 and it can also be partitioned into squares such that each one contains the numbers 1 to 9. This is illustrated in the following: each of the shaded regions contains exactly the numbers 1 to 9:

See also here for more information.

By putting a point in each small square where there is a 1 and leaving the other small squares empty we get the following point set:

This can be understood as a point set in which has the -net property in base 3 described above. This follows since the point set inherits the structure of the Sudoku.

**A higher order Sudoku**

The Sudoku presented above has a number of further properties, as illustrated in the following. Note that the shaded region again contains all the numbers from 1 to 9.

Thus, by replacing 1 with a point and leave the other areas empty, the point set above also inherits this structure. These additional conditions are similar to higher order nets.

**Higher order Sudoku puzzles**

In analogy to higher order nets, we call a Sudoku which satisfies all the constraints a higher order Sudoku.

In the following we present a (very difficult) higher order Sudoku puzzle. A printable pdf file can be found here.

That’s a nice connection to popular mathematics. Thank you!

(In the first paragraph, there is a minor typo: it should be “c+d=2”, not “c+d=1”.)

Thanks, I replaced the by .