Higher order nets and a higher order Sudoku

In the following we first describe a connection of $(t,m,s)$-nets and Sudoku and then generalize the Sudoku in analogy to higher order $(t,\alpha, \beta, m, n, s)$-nets in base 3. A new type of Sudoku puzzle is presented at the end.

A $(0,2,2)$-net in base 3 is a set of 9 points in the square $[0,1] \times [0,1]$ which have the following property:

Each rectangle

$\displaystyle \left[\frac{a}{3^{c}}, \frac{a+1}{3^c} \right) \times \left[\frac{b}{3^d}, \frac{b+1}{3^d} \right)$

for nonnegative integers $0 \le a < 3^c$, $0 \le b < 3^d$, $c + d = 2$, 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 $3 \times 3$ 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 $[0,1]\times [0,1]$ which has the $(0,2,2)$-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 $6 \times 9 = 54$ 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.

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2 responses to “Higher order nets and a higher order Sudoku”

1. Daniel G.

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 $1$ by $2$.