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Panorama

In the Atlantic Ocean there is an isolated island, that is in fact the top of a largely under water lying mountain. Of course this island is not very flat, it is full of peaks and low lands, valleys and plateaus. From places at higher altitude you have a nice overview over the island but from a valley not much more than the direct surroundings can be seen.

The numbers in the grid tell how many pieces of the island are visible from that cell in the six main directions. Only cells that are at the same altitude or lower are visible, and a cell at higher altitude blocks the rest of the line. Neighbouring cells can have a difference in altitude of at most 1 and the entire island is situated above sea level. The beach (all cells on the border or next to water) is entirely at altitude 1 (which is the lightest shade of colour available). We will solve a small puzzle of this type as an example.

Example

The cells at the borders all have altitude 1 (the lightest shade available), and when those have been coloured, only 7 cells remain. Let us first concentrate on the cell with the number 10 in it. When we count the number of cells in all directions, we learn that there are exactly 10 of those. This automatically means that the cell with the number 10 in it must be at a high enough altitude to see them all.

Now assume the cell with the number 10 in it has altitude 1, then all of the 10 cells that can be seen from there, must be at altitude 1. As a direct consequence the cell with the number 5 in it is now completely surrounded by cells at altitude 1, which means that (regardless the exact altitude of this cell itself) from that cell more cells can be seen than is allowed. Our assumption that the cell with the number 10 in it is at altitude 1 must be wrong. Because it borders to the beach, the actual altitude can be nothing but 2.

The cell with the number 5 in it, can see 3 cells at the beach for sure. Let us assume it has altitude 2 itself, then to the north 2 extra cells can be seen (because a cell bordering to the beach has at most altitude 2), and the same holds for the cells to the south east. Therefore the cell with the number 5 in it must have altitude 1.

All remaining cells border to a square with altitude 1, which means that in case the cell with the number 7 in it has altitude 2, it can see more than allowed, so it must have altitude 1. At the beach it can see 3 cells with absolute certainty, so there have to be exactly 4 more somewhere. Both to the north and to the south west it could see either 0 or 2 more. To the north west it could see either 0, 1 or 3 more. The only way to make exactly 4 from this, is by putting the cells to the north and to the south west at altitude 1 and the cell to the north west at altitude 2. To finish the puzzle we have to block the view to the north from the cell with the number 5 in it, by putting it at altitude 2.

Background

This genre was invented by Maarten Löffler.


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