Let us define a potential Kakuro grid as a finite rectangular lattice of white and black cells having the following properties:
- The grid is connected, i.e. every white cell can be reached from any other white cell via a finite sequence of single-cell horizontal and/or vertical moves.
- Every white cell has at least one immediate horizontal neighbor and at least one immediate vertical neighbor (i.e. no 1-digit words are permitted).
- Every row and every column contains at least one (and hence at least two) white cells.
- No word may be longer than 9 cells (or some other number, depending on what number system base is being used).
Finally, let us define a uniquely legitimate Kakuro grid as a legitimate grid for which there exists, up to isomorphism, only one legitimate sum set.
Now, I do believe that a 2x2 grid with no black cells is a uniquely legitimate Kakuro grid. Check me out on this, I'm not completely sure.
What other uniquely legitimate Kakuro grids are there? Is that 6x6 ATK grid, with a black cell in two corners and two black cells in the middle. a uniquely legitimate grid?
Gee, we could have our own Patterns Game conversation, just like the one up there in the Sudoku thread.