# permutations – Number of solutions and minimum cues in Sixy Sudoku

Sixy Sudoku is a variation on the Latin places and the traditional sudoku played on a $$6 times 6$$ grid with an initial index of several cells filled with a subset of digits $$1$$$$6$$. The task is to fill the remaining cells so that each number appears once in each

• $$1 times 6$$ row
• $$6 times 1$$ column
• $$2 times 3$$ shaded rectangle
• $$3 times 2$$ delimited rectangle

Questions

• Given a grid without initial filled cells, how many valid filled grids, $$K$$, is there (up to the symmetry of permutation of numbers)?
• What is the minimum number of cells filled, $$n ^ *$$, which guarantees a unique solution?
• For this minimum $$n ^ *$$, how many distinct cell locations guarantee a unique solution (up to the permutation symmetry of digits)?

For the first problem, without loss of generality, we can define the numbers in the shaded rectangle at the top left, as shown below:

A naive upper limit on the number of valid filled grids, $$K$$consists in considering each of the remaining shaded rectangles as independent, which gives $$(6!) ^ 5$$ solutions. (The same logic applies to the consideration of independent lines, or independent columns, or separate rectangles.) Of course, this limit will be extremely loose because it will not incorporate many constraints.

A slightly narrower limit can be found by considering the left set of three shaded rectangles as independent, and then adding the row constraint for each shaded rectangle aligned to the left. That way we get $$(6!) ^ 2 ((3!) ^ 2) ^ 3$$. But of course, this bound does not include any constraints, such as column constraint. A narrow limit on the number of bits needed to specify a problem (initial cell specifications) with a single solution is $$log_2 K$$.

For the last two problems, it will be interesting to see the proximity of the information defined by the minimum number of cells filled, $$n ^ *$$ (or $$n ^ * geq 5$$ for the specification of figures), and the candidate placements approximate the related information given by $$K$$.