Long ago, on the planet Sudonia, the Sudonites were ruled by a vengeful, but infinitely logical god, Spuck. Actually, unbenknownst to the Sudonians, Spuck was an avatar of an advanced alien race, the Valcons who had discovered Sudonia and taken over by pretending to be a god. But to the Sudonians, it was all about the same.
Now, because Spuck was infinitely logical, he was exceptionally fond of solving sudoku puzzles and required of his followers (ie the Sudonians) that the puzzle and it's logical ideals permeate everything that they did, and every part of their culture. As the Sudokians became more technologically advanced, they began to industrialize and build cities. Spuck decreed that all cities must be 9 blocks by 9 blocks, each block containing one building which must be less than 9 stories, each row, column and three by three square must contain 1 building of each size (you get the idea), and then enough of the buildings must contain their height inscribed on the roof top so that when Spuck looked down from heaven (ie his space ship) he could logically deduce the heights of the rest of the buildings. (The Valcons actually required this so that they could buzz the towns in their invisible hover-cars without having to worry about hitting a building). This seemed a little bizarre to the Sudonians, but they went along with it. Those architects who did not conform to Spuck's rules (especially the solvability one) were smited from heaven by lightning and never heard from again (I did mention his vengefulness).
After a few hundred years, however, the Sudonian city-architects began to get a little disgruntled with this arrangements. They, or rather, a small minority of artistically inclined architects, felt that the requirements were stiffling their creativity, and besides, all those numbers on the top were hard to maintain. So they prayed to Spuck and asked for a special dispensation, in the name of Art, to allow them to build a small number of towns using different rules. Spuck relented and decreed that new cities could be built with arrows on the buildings which would indicate that the height of the building was the same as the number of other buildings "visible" from that building in that direction (ala the rules of Hidden Skyscraper Sudoku). Of course, the heights of all buildings in the city would have to be deducable from the locations of the arrows. This made the Sudonians happy, and they built several cities to these specifications (and those who didn't were fried, and their beautiful cities destroyed).
Now, all the Valcons buzzing the cities of Sudonia made for a lot of noise which disturbed the Sudonians in their logical meditations, and so they prayed to Spuck to make the noises stop. But Spuck said "Lo, those noises are my displeasure with the ease of solving the puzzles that your cities pose for me. If you would have them end, then you must create for me a more difficult puzzle." Unfortunately, this was not possible as Spuck was infinitely logical (and anyway, the Valcons weren't going to give up their joy ridings).
At about this time, a group of Sudonians decided to build a city near one of the coasts in a temperate climate, and commissioned the most famous city-architect of the day, Froyd Lightwank to design the layout of the buildings. They implored him to try his best to create a puzzle which would put an end to Spuck's grumblings, and he promised them that he would do everything he could. For many months Froyd toiled to create the ultimate HSS puzzle that would have only one solution but would be too difficult for even a god to solve. And of course, he was doomed to failure. He started to despair that he wouldn't be able to realize the city-founder's dreams of a peaceful metropolis. But, one cool morning, while looking out over the bay, he had an epiphany. He ran back to his office and drew up plans for the city. The construction crews set to work immediately and created the following map (as viewed from above):
Everybody agreed that this was an exceptionally difficult HSS puzzle, and while nobody was able to solve it, they were assured by Froyd that there was in fact a solution, but even Spuck would be hard pressed to find it. So they went ahead and built the city.
On a beautiful sunny day, as the last building was completed, the founders had a dedication ceremony to honor Froyd for his creation. As the guests were dancing and enjoying their cakes, one humble little girl named Motria who was Froyd's favorite neice, rushed up to the mayor and Froyd and said: "Um, Mr. Mayor, Uncle Froyd, I think that this city is doomed." The mayor was shocked "Why, whatever do you mean, little girl? The city is just fine. Run along now and play with your friends, and leave the adults to worry about such things." But Motria persisted. "But I think there are at least two configurations that satisfy the arrow diagram. In addition to the building configuration (which we can see from the side up here on the hill), I can switch these 4 buildings around (she pointed to a map indicating which buildings) creating another valid configuration." Now that a solution to the puzzle was apparant, it was not difficult to find extra solutions to it which had eluded even the best HSS solvers until now. The mayor was appalled. By now, several other guests had started to gather around the three of them. After Motria finished her explanation, everybody gasped, and, except the little girl, took several steps back from Froyd (afterall, while you wouldn't wish for your neighbor to be smited, you wouldn't want to be standing next to them when they were).
But Froyd smiled and said, "You're a very smart Motria, but I'm not worried." And indeed, Froyd was still standing and not smited. "Please, Motria (and anybody else who is worried), stay here on this hill tonight and tell me in the morning if you're not satisfied. I'm tired, now, though, so I think I'll go get some rest." And he walked down into the town and slept the sleep of the innocent. The rest of the city, however, were still too worried to sleep in the town lest they be smited along with the buildings, and stayed on the hill all night.
Early the next morning, a bitter chill had set in over the night. As the sun rose, the people of the town looked out over a very different city. You see, a fog had rolled in, and covered the buildings up to a certain level. As the fog rolled in, the arrows on top of the buildings began to change (they were automatically set to self-configure by measuring their vertical distance to the ground, and by viewing other buildings in each direction. With the fog rolling in, they were getting different readings because their apparent heights were altered, and many buildings were completely concealed by the fog, thus changing the number of visible buildings). When the fog had settled into a nice even level, the new map of the city looked like this:
Just then, Froyd appeared out of the fog with a big smile on his face. Motria jumped up into her uncle's arms and said, "You're the smartest architect ever. With the new map, I can deduce not only the height of the fog, but also the heights of the buildings perfectly."
Now, Spuck (or rather the Valcons) could see this also, but much to their horror, they realized also that they couldn't buzz the town because they would never see both maps at the same time, and thus never be able to correctly deduce the heights of the buildings. The town was left in peace for many millenia, and Froyd and his neice (who grew up to be an even more successful architect than her uncle) are honored to this day as heroes.
Ok, so the rules:
The first grid is a standard Hidden Skyscraper Sudoku:
Fill in the grid with numbers 1-9 so that each row, column, and 3x3 square has 1 of each digit. The arrows mean that the height of the building on that square correctly indicates the number of visible buildings in that direction. A lack of an arrow on a square means that the height of that building INCORRECTLY indicates the number of visible buildings in that direction.
The second grid is a little different.
Let H be the height (in stories) of the fog. Each square in the second grid is H less than the corresponding number in the first grid (or 0 if the building on that square is H stories or less, ie it is completely covered by the fog). For example, if the fog was 3 stories thick, then all 4's in the original grid are 1's in the second grid, all 9's are now 6's, and 1's, 2's and 3's are 0's.
Again, an arrow in a square means that the height of that building ABOVE the fog correctly indicates the number of buildings which can be seen in that direction. Any building that is completely covered in fog is not included in the counts.
A missing arrow indicates either that the building on that square is completely covered in fog, OR that the height above the fog does NOT accurately indicate the number of buildings visible (over the fog) in that direction.
