Over in the djape forum, my friend HATMAN and I are discussing about the possibility of a non-consecutive Windoku (or NRC-Sudoku, see this link). By "non-consecutive" we mean that all horizontally/vertically adjacent neighbours cannot contain consecutive numbers (such as "23" or "98"). 1 & 9 are not considered consecutive so are allowed to be adjacent to each other.
To our surprise we could not find any Windoku solution grid being non-consecutive. Not even one with only 3 of the 4 "windows" as constraints (a window is one of r234c234, r234c678, r678c234, r678c678). HATMAN did find a grid with 2 windows side by side that allow a non-consecutive solution grid (as a result he even created 2 puzzles with only 1 killer-cage, which must be a record so far)...
This prompt us to the following conjecture:
The Non-consecutive Windoku Conjecture
For all solution grids, if you require the following three 9-cell groups to have no repeating:
r234c234, r234c678, r678c234
... then there must be a least one pair of adjacent neighbours containing consecutive values.
It is obviously too hard to prove this conjecture manually. So I hope the programming masters can help proving it using brute force. I think it wouldn't be too hard if you set the 3 windows aforementioned as additional constraints, and doing a forward check search with the non-consecutiveness as additional elimination rules. What we want to know is if it really is true that 3 windows must force consecutive neighbours.
I hope high-calibre programmers interested in solution grid structure (e.g. gsf, Red Ed, coloin and many others) could lend a hand here. Thanks in advance!
