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Problem of the Week 920

Plane Color

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Given n doubly-infinite lines in the plane, one can color the regions into which they divide the plane in alternating fashion, black and white. Define the discrepancy d as the absolute value of the difference between the number of black regions and the number of white regions.

Arrange some lines in the plane so that the discrepancy is at least 2.

How large a discrepancy can you get?

Example: The following arrangement has discrepancy 1

          |     |
       B  |  W  |  B
          |     |
     -----|-----|------
          |     |
       W  |  B  |  W
          |     |
     -----|-----|------
          |     |
       B  |  W  |  B
          |     |
Source: Jim Propp (U.Wisconsin-Madison)
© Copyright 2000 Stan Wagon. Reproduced with permission.

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25 October 2000