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Problem of the Month (August 2014)

Suppose we have n points in the interior of a square, and these points are rigid because they are constrained to be distance 1 from some subset of the sides of the square and each other. What are the possible sizes of the square, and the fewest number of points necessary to achieve them? The configurations for n=1 and n=2 are shown below.


2

2 - 1/√2 = 1.292+

2 + 1/√2 = 2.707+

3

What if the points are constrained to be distance 1 from some subset of the corners of the square and each other?


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