Are there any valid continuous Sudoku grids?

A standard Sudoku is a 9×9 grid filled with digits such that every row, column, and 3×3 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 3×3 box restriction in standard Sudoku.

Let U=[0,1] and U2=U×U. More precisely, a weak continuous Sudoku is essentially a function f:U2U, which satisfies the following four properties:

  1. If x,y1,y2U and y1y2, then f(x,y1)f(x,y2).
  2. If x1,x2,yU and x1x2, then f(x1,y)f(x2,y).
  3. If xU then {z:f(x,y)=z,yU}=U.
  4. If yU then {z:f(x,y)=z,xU}=U.

Now, strong continuous Sudoku is a bit harder to define precisely. A set S is a square sub-region of U2 iff SU2 and there exists z=(z1,z2)U2 and r>0 such that S={(x,y)U2:z1xz1+r,z2yz2+r}. Thus, using this definition, a strong continuous Sudoku is a weak continuous Sudoku which satisfies the following additional property:

  1. If S is a square sub-region of U2, 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)={x+yif x+y1x+y1if x+y>1
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?

Answer

Weak continuous Sudoku:

A weak continuous Sudoku can be constructed based on the ideas that you already provided.

First, we construct a weak continuous Sudoku for the set U=(0,1] instead of U=[0,1].
Here, a weak continuous Sudoku can be constructed by using the function
f from your attempt but as a function f:(0,1]2(0,1] (since one boundary is gone, the problems that you observed are now gone, too).
Then, choose a bijection h:[0,1](0,1]
(an explicit bijection can be constructed if you prefer a constructive soution).
Then we define
g:[0,1]2[0,1],(x,y)h1(f(h(x),h(y))).
This function g then can be shown to be a weak continuous Sudoku.

Strong continuous Sudoku:

As for strong continuous Sudoku, things get more complicated
and it would be a lot of work to explain my construction in full detail,
but I can provide a sketch.

First, the bijection h above should be chosen such that
each interval in [0,1] contains a subinterval [a,b] such that h(x)=x
for all x[a,b], see the comments below for such a construction.
Furthermore, it uses a bijection j:[0,1][0,1]
such that j((c,d)) is dense in [0,1] for all intervals (c,d),
see the comments below for such a construction for j.

Then one can mix the rows or columns of the previous weak Sudoku according to j,
i.e. ˜g(x,y)=g(j(x),y).
This function ˜g should then be a strong continuous Sudoku.
Let me provide a rough sketch how this can be done.

Let S be a square sub-region of [0,1]2.
Let S2=[a,b]×[c,d]S be a smaller square sub-region,
where a<b,c<d are such that
h(x)=x holds for all x[a,b][c,d]
(such a sub-region exists due to the comments above on the choice of h).
It suffices to show that ˜g(S2)=[0,1] instead of ˜g(S)=[0,1].

Let t[0,1] be given.
Let m:=(c+d)/2.
Since j([a,b]) is dense in [0,1],
the function values {˜g(x,m)|x[a,b]} are also dense in [0,1].
Let s[a,b] be such that ˜g(s,m) is close to t in the sense that
tdc2<˜g(s,m)<t+dc2.
By exploiting the definitions of ˜g,g,f we have
˜g(s,m+x)=˜g(s,m)+x
for x(dc2,dc2)
(with the exception that the values wrap around at 1).
By setting x=˜g(s,m)t,
we get t=˜g(s,m+x) and (s,m+x)S2=[a,b]×[c,d].
Thus t can be reached and the condition (5.) for strong continuous Sudoku
is satisfied.

on the existence of a function h:

We can define h:[0,1](0,1] by setting h(0)=1/2, h(1/2)=1/3, h(1/3)=1/4,
etc., and h(x)=x for all other x.
Then for each interval one can find a sufficiently small subinterval
[a,b] such that h(x)=x for all x[a,b].

on the existence of a function j:

This is more complicated, so let me provide a rough sketch.
Let (qk)k be an enumeration of the rational numbers in [0,1]
and let Ik be an interval of length 232k centered at qk.
We define the sets
Ak:=Ikl>kIl.
These sets form a partition of [0,1] and each set Ak has cardinality
equal to [0,1].

Let (Bk)k be another sequence of subsets of [0,1] which
form a partition of [0,1] such that each Bk is dense and has
cardinality equal to [0,1]
(such a partition exists, one can append dense countable sets with enough other elements to form sets Bk,
but I think this requires the axiom of choice).
Then we construct j by (bijectively) mapping
Ak to Bk.

Since the lengths of the sets Ak get smaller and smaller
and the rationals qk are dense,
each interval has a subinterval of the form Ik.
Since Ik contains Ak and Ak is mapped to a dense set Bk,
we obtain the desired property that j(Ik) is dense in [0,1].

Attribution
Source : Link , Question Author : ZKG , Answer Author : supinf

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