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Rectangles

A farmer wants to divide his land into nice rectangular pieces to give to his sons. However, he doesn't like all of his sons equally much, so he wants to give each son only a certain number of fertile land cells. Obviously, lakes aren't fertile. Right?

Place fences to divide the farmland into rectangles. Every rectangle should contain exactly one son (indicated by a number), and the number should equal the number of land (not water) cells in the rectangle. Fences cannot be placed on water.

Example

In this example, the top 4 needs to be in a rectangle that has four non-water squares, and contains no other numbers. This is only possible in one way: the top row has only three non-water squares, so the cell below the 4 needs to be part of the rectangle. Now, if it would contain the lake next to the 4, the area can only be 3 or 5. It cannot be a narrow rectangle going only down either. So, it has to be a square.

Now that the top left square has been filled, the rest follows easily. The tobbom left 4 has to be a 4x1 rectangle, which cuts the rest of the diagram into the required areas of area 3.


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