Beacons and Towers
(Published on 18. September 2022, 21:21 by Elliptical)
RULES
Chaos Construction: Divide the grid into nine orthogonally connected 9-cell large non-overlapping regions. Place the digits 1-9 once each in every row, column, and region.
Watchtowers: A cell is a watchtower if its digit is equal to the number of cells it can see along its row and column combined (including the cell itself which counts once to that total). Region borders block the watchtower’s view. The puzzle does not provide the location of any watchtowers. The solver must use the beacons to infer the location of some of the watchtowers.
Beacons: Cells with arrows are beacons. The digit in a beacon indicates the distance to a watchtower in the direction of the arrow. That watchtower is not necessarily the first watchtower in that direction. A beacon with multiple arrows points to multiple watchtowers. A beacon and all the watchtowers it points to must all be in the same region. A cell may be both a watchtower and a beacon simultaneously.
There are no negative constraints: Not all watchtowers must necessarily have beacon(s) pointing to them and not all possible beacons are necessarily given.
This is a 5x5 example to help illustrate the rules.
I appreciate solves, comments, and suggestions
Solve the puzzle at Penpa+ with answer check. To trigger the answer check you must fill all the digits and delineate the region borders using green edges.
I estimate the difficulty level of that puzzle to be somewhere between 3 and 4 starts out of 5.
Solution code: Row 6 (left to right) using dashes ‘-’ for region borders, followed by an equal sign ‘=’, followed by column 4 (top to bottom) using dashes ‘-’ for region borders. It should be something like 123-4-567-89=9-8765-43-21.
Comments
on 22. November 2022, 12:47 by ONeill
Very nice! An impressive construction, thanks :)
on 21. September 2022, 22:16 by marcmees
very rewarding solve. Hard break in. Very nice. thanks.
on 20. September 2022, 15:10 by PrimeWeasel
Of all the possible ways to break this puzzle, I'm proud to say, I have found them all.
on 20. September 2022, 12:29 by Piatato
Very fascinating! Great construction!
on 19. September 2022, 23:18 by lerroyy
Very nice puzzle! thanks
on 19. September 2022, 16:53 by TotallyNormalCat
Amazing puzzle, I had a lot of fun even though I broke it at least 2 times :P
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Thanks for the kind words. Don't worry about breaking the puzzle. I've broken some of your puzzles in the past as well :(, but always find them very enjoyable :).
on 19. September 2022, 16:07 by filuta
I found start very elegant and not too hard (once I read the rules carefully enough - which took me some time admittedly), the hardest part for me was the end which makes the puzzle 4 star in my eyes. the clues are chosen very carefully, which makes the puzzle blossom like a flower - quite miraculous.
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Thanks for the solve and the comment. I really appreciate it. I appreciate in particular the feedback on what part of the solution you found hardest to solve. That's invaluable information that will help me set better puzzles in the future.
on 19. September 2022, 12:45 by lerroyy
Hi, I have 3 questions
Do the cells between watchtowers and the beacon that point to it have to be in the same region?
Can a watchtower be on a beacon cell who points to another watchtower?
For beacons with multiple arrows, is the digit on the beacon = to the sum of the distance between it and watchtower 1 and it with watchtower 2 or are both watchtowers at the same distance of the beacon, and it is the distance to 1 of the watchtowers?
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Answer 1: Cells between a beacon and its watchtower may belong to a different region (or multiple different regions) All you know is that the beacon and its tower(s) are in the same region.
Answer 2: Yes, a cell can be simultaneously a beacon and a watchtower in which case it has a beacon pointing to it and it is also pointing to another watchtower.
Answer 3: For a beacon with two arrows, both watchtowers must be at the same distance to the beacon and that distance is the digit in the beacon cell.
on 18. September 2022, 21:26 by Elliptical
Removed some minor typos
