Number of solutions and minimal clues in Sixy Sudoku 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



Questions


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*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:

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:

(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 $K$.

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:

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.
 A: As for the first question, a backtracking algorithm, see sixy.c at
https://github.com/wilberdk/sixy
shows there are 1936 completions of 
$$\matrix{1&2&3&4&5&6\cr
*&*&*&*&*& *\cr
*&*&*&*&*&*\cr
*&*&*&*&*&*\cr
*&*&*&*&*&*\cr
*&*&*&*&*&*&}$$
The answer to the second question is $n^*=7$. 
The answer to the third question is $908\,928$.
These are more tricky. For instance, 
there are 2752 ways to chose five cells in
$$\matrix{
*&*&*&*&*& *\cr
*&*&*&*&*& *\cr
*&*&6&*&*&*\cr
*&*&*&*&*&*\cr
*&*&*&*&6&*\cr
*&*&*&*&*&*}$$
to construct an initial clue like
$$\matrix{
*&*&*&*&*& *\cr
*&*&*&*&*& 1\cr
*&*&6&*&2&*\cr
*&3&*&*&*&*\cr
*&4&*&*&6&5\cr
*&*&*&*&*&*}$$
that ensures a unique solution.
Here the construction fills the chosen cells with 1 through 5 consistent with the lexicographic order.
One sees that $n^*=7$ by replacing 
$$\matrix{
*&*&*&*&*& *\cr
*&*&*&*&*& *\cr
*&*&6&*&*&*\cr
*&*&*&*&*&*\cr
*&*&*&*&6&*\cr
*&*&*&*&*&*}$$
with  starting configurations that have just one filled cell. 
There is a symmetry group $G$ of order 128 with just three orbits of cells, and we tried all 6 times 3 orbits of 
configurations that have just one filled cell. The group $G$ is generated by the following three operations: 
turn the grid upside down; reflect the grid in the main diagonal; interchange the first two rows.
To be more specific about the answer to the third question, 
let us  make some definitions.
Call a clue valid, if it has seven filled cells and a unique solution.
Given a valid clue, its core is obtained by deleting all integers that occur only once.
The core size of a valid clue is the number of filled cells in its core.
For instance the core size of 
$$\matrix{
*&*&*&*&*& *\cr
*&*&*&*&*& 1\cr
*&*&6&*&2&*\cr
*&3&*&*&*&*\cr
*&4&*&*&6&5\cr
*&*&*&*&*&*}$$
is two.
Possible core sizes are two, three and four.
One tries all $G$-orbits of possible cores.
There are $6!$ times $396\,800$ valid clues with core size 2 and they form $6!$ times $12\,242$ $G$-orbits.
There are $6!$ times $16\,000$ valid clues with core size 3 and they form $6!$ times $226$ $G$-orbits.
There are $6!$ times $496\,128$ valid clues with core size 4 and they form $6!/2$ times $5320$ orbits under a group which is twice as large as $G$.
$396800 + 16000 + 496128 = 908928$.
The programs we wrote for this task can be found at 
https://github.com/wilberdk/sixy
