I don't know whether koushanejad74 is aware of this, or anyone else has noticed it, but apart from r46c46, r5c5, the only values that can possibly go into a grey cell in a Benoku puzzle are 1 and 9.
In r5c5, 5 can also go there. In r46c6 4 and 6 can also go there.
The reason for this is easy to see. The grey cells are the only ones allowed in their row and column in a Benuko puzzle.
If the grey cell is on any edge cell of the puzzle, it sees an 8 cell compartment in its row or column, which must have values 1-8 or 2-9, so the grey cell can only be 1 or 9.
If the grey cell is in Row 5 or Column 5 (but not in r5c5), it splits the column or row into two 4 cell compartments and can only have the values 1, 5, or 9. However, in the row or column there is one compartment that contains >= 5 cells, which must contain a 5, which leaves only 1 and 9 for the grey cell.
Now look at off diagonal cells, for example r4c3. A little bit of thought will lead to the conclusion that the grey cell can only have the values 1, 3, 7 or 9 considering the row (which must contain one 2 cell compartment and one 6 cell compartment) or 1, 4, 6 and 9 considering the column (which must contain one 3 cell compartment and one 5 cell compartment). Since it can't be 3, 4, 6, or 7 at the same time the only allowed values are 1 and 9.
So far, everything I've said is valid for any Str8ts puzzle (which also allows blanks in grey cells, but Benoku does not).
Now look at a diagonal cell, (other than Box 5). Pick, say r3c3 for example. The allowed values are 1, 3, 7 or 9 for both the row and the column. So you might conclude that it could be 3 or 7. This is where the boxes come in. r3c3 is in Box 1 and the 2 cell compartments in the grey cell's row and column must contain different values, so the grey cell can't be 3 or 7 at the same time, so again it can only be 1 or 9. A similar argument can be made for any off-diagonal cell (I think).
In r5c5. The allowed values are 1, 9 and 5, and 5 does not cause any sort of contradiction of the sort described above. Similarly in r46c6 you can have 1, 4, 6 and 9 and this does not cause a contradiction. QED (hopefully).
This fact greatly reduces the scope for finding puzzles in Benoku. The best way to find a puzzle is to pick your grey cell positions, put 1 or 9 in them (or 5 in r5c5, 4 or 6 in r46c46) and test for a contradiction (not all 1/9/5/46 grey cell patterns are mutually compatible). If no contradiction is found then there are usually lots of possible puzzles, which you can generate by filling in clues until you get a unique solution. With luck you can then also remove some or all of the 1's and 9's (and 5/46) from the grey cells and still have a unique solution puzzle. At least that's what I've been doing.
I'll finish off by presenting just one more puzzle, which has three different values in grey cells, and is not trivial to solve.
- Code: Select all
0 0 0 1 0 0 0 0 0
0 0 7 0 0 0 0 [9] 0
0 0 0 2 0 3 0 0 0
8 0 0 0 0 0 0 5 0
0 0 0 0 [5] 0 0 0 0
0 0 3 0 0 0 9 0 0
0 [1] 0 0 0 7 2 4 0
0 0 0 0 0 0 0 0 5
0 0 4 0 0 0 0 0 0
Leren
PS I thought of the 46 in r46c6 thing at the last minute and tested it. It works.
