A standard Sudoku is a $9\times 9$ grid filled with digits such that every row, column, and $3\times 3$ box contains all the integers from $1$ to $9$.
I am thinking about a generalization of Sudoku which I call "continuous Sudoku", which consists of a unit square where every point on that square corresponds to a real number. The rules for continuous Sudoku are designed to be analogous to the rules for standard Sudoku, and I've devised two different rulesets:
- The first ruleset I call "weak" continuous Sudoku. In weak continuous Sudoku, the only restriction is that every row and column of the square contains every real number in the interval $[0,1]$ exactly once.
- The second ruleset I call "strong" continuous Sudoku. In strong continuous Sudoku, the rules of weak continuous Sudoku apply, and, in addition, every square sub-region of the unit square contains every real number in the interval $[0,1]$ at least once. This is analogous to the $3\times 3$ box restriction in standard Sudoku.
Let $U = [0,1]$ and $U^2 = U\times U$. More precisely, a weak continuous Sudoku is essentially a function $f:U^2\to U$, which satisfies the following four properties:
- If $x,y_1,y_2\in U$ and $y_1\neq y_2$, then $f(x,y_1)\neq f(x,y_2)$.
- If $x_1,x_2,y\in U$ and $x_1\neq x_2$, then $f(x_1,y)\neq f(x_2,y)$.
- If $x\in U$ then $\{z: f(x,y)=z,y\in U\} = U$.
- If $y\in U$ then $\{z: f(x,y)=z,x\in U\} = U$.
Now, strong continuous Sudoku is a bit harder to define precisely. A set $S$ is a square sub-region of $U^2$ iff $S\subseteq U^2$ and there exists $z = (z_1,z_2)\in U^2$ and $r>0$ such that $S = \{(x,y)\in U^2:z_1\leq x\leq z_1+r,z_2\leq y\leq z_2+r\}$. Thus, using this definition, a strong continuous Sudoku is a weak continuous Sudoku which satisfies the following additional property:
- If $S$ is a square sub-region of $U^2$, then $f(S) = U$.
I've been trying to look for specific examples of both strong and weak continuous Sudoku grids, but have so far been unsucessful.
I'm not sure whether any weak continuous Sudoku exists. My first attempt: $$ f(x,y)=\begin{cases} x+y &\text{if }x+y\leq 1 \\ x+y-1 & \text{if }x+y>1\end{cases} $$ almost works. It satisfies properties $3$ and $4$, and almost, but not quite, satisfies $1$ and $2$. The issue occurs only at boundaries of the square, for example, $f(0.5,0) = 0.5$ and $f(0.5,1)=0.5$.
Any example of a strong continuous Sudoku will likely need to be an extremely discontinuous pathological function, similar to the Conway base 13 function. Obviously, if there are no weak continuous Sudoku grids, then there are no strong continuous Sudoku grids. Even if there are no weak Sudoku grids, it may be possible to slightly modify the definitions to permit small exceptions such as in the above example.
The main question I'm asking is: Do any weak continuous Sudoku grids exist, and if they do, do any strong continuous Sudoku grids exist?
from Hot Weekly Questions - Mathematics Stack Exchange
ZKG
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