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A serpentine is a sequence of points

P_1 , ..., P_m

in a plane, not necessarily all different, such that the distance between

P_i

and

P_{i+1}

is at least 1, and the segments

P_i P_{i +1}

are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine (

P_m = P_i

for some i < m).

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Problems(1)