The New Sudoku Players' Forum

[Mathimagics]

I've been investigating what limits there are on the number of blank cells (ie: cells which contain numbers), for grids of various sizes, on which a well-formed puzzle can be set, ie a puzzle with one unique solution.

Consider this example:

Nearly symmetric grid, size 15, unique solution

KP15b_NH45.jpg (113.17 KiB) Viewed 1498 times

This represents an attempt to create a unique solution on a 15x15 grid (ie. 14x14 working area) with dual symmetry, having 152 blank cells, and 44 hint cells (excluding the zero/top row and zero/left column).

But as you can see, it is in fact one blank cell short of the target (the cell at row 6, col 8, is a hint cell with value "0\15").

Now I'd like to change that cell to a blank. The original solution allows two values to be set in this position, so I adjust the two hints that changing this cell affects, ie at (0,8) and (6,4), to reflect the chosen value, and hey, presto ... the puzzle not only loses uniqueness of solution, it just gets completely blown away.

The first value chosen led to 1.9 million solutions, and the second has 4 million solutions (and in fact is still running)!

[Mathimagics]

Here is the solution to that puzzle:

KP15b_NH45_Soln 24.jpg (199.3 KiB) Viewed 1496 times

As you can see, I can place a 5 or a 9 at cell (6, 8) so the adjusted sums become (0, 8) = 21 or 25, and (6, 4) = 26 or 30.

The problem I think stems from the horizontal word at (6,4). By merging it with the word at (6,9), the sum created allows many more possible settings, each of which ripples through the rest of the puzzle, creating more and more ambiguity.

BTW, the second setting (6,8) = 9 came in at 12.7 million solutions