IBM Ponder This #067 · November 2003

This puzzle is adapted from one published by S. B. Gashkov. The updated version is due to Sasha Ravsky, Gyozo Nagy, Gabor Sagi and the puzzlemaster.

Give the exact value for s, and an approximation correct to three decimal places, making the following statements true; provide justification for both the upper and lower bounds.

P is a regular pentagon one meter on each side. Consider a curve C of
some specified length T, lying within P, containing no straight line segments.
For any curve whose length T exceeds s, there is a straight line that intersects
the curve at least six times. But for any T less than s, there is a curve of
length T such that any straight line hits it at most five times.

Solution

Best opened after a real attempt

To be added.