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[The 1000 point problem on SRM 209, Div I]

At some stage, the problem reduces to the following:

Given blocks of three square units, like below, which can be rotated in any manner, how many ways are there to fill a rectangular block of given size.

| x | x |
| x |

For example, for a block of 3x4, there are 4 ways of arranging these blocks. I am looking for a way to attack this problem, and not the actual solution. How do I go about finding the number of ways. There are so many ways that it could happen, and I do not see overlapping sub problems for a DP approach either.

Any insights are welcome.

Moeb
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1 Answers1

-1

Without exception, every tiling of a pxq block of space with L-shaped tiles, will reduce to tiling with 2x3 blocks consisting pairs of your L-shaped tiled. I.e. the tiles are either in the form:

        xx      xx
        xy  or  yx  to form a vertical 2x3 block or
        yy      yy

        xyy       xxy
        xxy  or   xyy  to form a horizontal 3,2 block.

So you can already reduce your problem to a 'parquet'-tiling of a rectangle with 2x3 and 3x2 rectangles. Unless, of course, you are tiling an irregular non-rectangular region - in which case you have to consider the L-shaped tiles individualy.

Penguino
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