**Sixy Sudoku** is a variation on Latin squares and traditional sudoku played on a $6 \times 6$ grid with an initial clue of several cells filled in with a subset of the digits $1$–$6$.  The task is to fill in the remaining cells such that each digit appears once in each

 - $1 \times 6$ row
 - $6 \times 1$ column
 - $2 \times 3$ shaded rectangle
 - $3 \times 2$ outlined rectangle

[![enter image description here][1]][1]

**Questions**

 - Given a grid with no initial filled cells, how many valid filled grids, $K$, exist (up to digit-permutation symmetry)?
 - What is the minimum number of filled cells, $n^*$, that guarantees a unique puzzle solution?  
 - For that minimum $n^*$, how many distinct placements of filled cells ensure a unique solution (up to digit-permutation symmetry)?

For the first problem, without loss of generality, we can set the digits in the upper-left shaded rectangle as shown here:

[![enter image description here][2]][2]

Guided by @GerardPaseman (below), we can see that there are $2^7$ ways to fill the top half of the puzzle (given the assigned upper-left shaded rectangle):  The top row has $2^2$ alternatives (given all constraints), and the second row has $2^2$ alternatives.  The third row has $2^3$ alternatives.  Putting together:  $2^7$.  But then there are the cells in the bottom half of the puzzle.  The naive (but slightly tighter) bound is thus $(2^7)^2$.  

The red show the number of independent alternatives in some of the cells starting at the right top shaded rectangle, then beneath it, then moving to the left:

[![enter image description here][3]][3]

(The blue arrows show the sequence of cell fillings using all prior constraints.  One can work in a different sequence of cell constraints, if desired.)

For the last two problems it will be interesting to see how close the information defined by the number of minimal filled cells, $n^*$ (where $n^* \geq 5$ for digit specification), and candidate placements approximates the information bound given by $K$.


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**Addendum**:

Today's **Sixy Sudoku** puzzle in *The New York Times magazine* is a variation on the traditional puzzle in which the bordered rectangles are replaced by $6$-element locking polyominoes, as shown:

[![5-clue puzzle][4]][4]

I confirmed that this puzzle is solvable.  Note that it has just $n=5$ clue cells.  This *suggests* (but of course doesn't prove) that $n^* = 5$ for the traditional rectangular Sixy Sudoku.

  [1]: https://i.sstatic.net/TWAEA.png
  [2]: https://i.sstatic.net/IBXpO.png
  [3]: https://i.sstatic.net/7jvpJ.png
  [4]: https://i.sstatic.net/2aWxR.png