Count the Borders
(Published on 21. January 2023, 20:18 by Tobias Brixner)
Background: In some difficult puzzles, the break-in is particularly complicated but afterwards the solve path is more straightforward. With this new set of rules, I have tried to make the start not too hard, and then to maintain a variety of types of logic. I hope it is fun to disover the regions and I would love to see your comments!
Puzzle links: Play on CTC SudokuPad (with solution check) or f-puzzles.
Rules: Place the numbers 1-9 once each into each row, column, and region of orthogonally connected cells. Regions must be determined. Cells connected by an "S" belong to the same region.
A digit in a cage counts the number of borders between regions as seen from that cell along orthogonal directions (up, down, left, and right) all the way out to the edge of the grid, with the cage clue indicating over how many of the four directions the borders are summed up. Only borders between regions are counted and not the outer edge of the grid. Relevant directions may vary from cage to cage. The "1&4" acts as both a 1-clue and a 4-clue.
Digits must not repeat in cages containing the same clue number. E.g., if there is a 7 in a 3-clue cage, there is no second 7 in any other 3-clue cage, but there may be a 7 in one of the other cage types.
Example: In the following partially filled grid, digits in cages are assigned. Pencil marks indicate that there is more than one possibility, depending on which directions are considered. A unique solution then requires further logic.
Let us first consider R3C2. Here the boundaries between regions have to be counted because of the hint for two of the four orthogonal directions (U=up, D=down, L=left, R=right). This results in the following possible combinations: UL=2, UR=7, UD=5, RD=8, RL=5, DL=3, and analogously for the other cage cells.
In R1C9, both L and D give the same count of 2, and the counts of 0 from U or R do not provide a valid Sudoku digit. Considering border counts alone, the 5 in R8C6 might also have been a 2, but that 2 is excluded because the rules do not allow doubling a digit in the same cage-clue type, and there already exists a 2 in R1C9. In R8C7, 5 might have been a third option, but it is excluded by Sudoku from the neighboring cell. In R7C9, 2 as a digit (for UR or UD) is excluded by Sudoku from R1C9, and 5 (from UL) is excluded because the cell sees another 5 in its region.
Solution code: All digits of row 9 (from left to right) followed by column 7 (from top to bottom) without spaces
Comments
on 20. May 2024, 15:54 by sanabas
Fantastic puzzle with unique logic
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Thanks a lot. Glad you liked it! - TB
on 5. November 2023, 08:36 by Lavender Gooms
Very fun indeed. Thank you!
on 17. October 2023, 13:10 by marcmees
One of the nicest chaos constructions made so far. challenging till the end. thanks
on 20. February 2023, 15:00 by Tobias Brixner
Thank you so much for playing and your very kind words, I am glad you liked it!
on 18. February 2023, 03:50 by Gnosis66
Great puzzle! I had to restart 3 times before I cracked it. It’s my favorite of the solves I’ve done in 2023.
on 22. January 2023, 02:02 by Myxo
Extremely hard and satisfying chaos construction. I highly recommend it!
