A toy wind-up mouse is put in a circle pen with 1 m radius.
geometry/mouse.in.pen New
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Problem
A toy wind-up mouse is put in a circle pen with 1 m radius. The mouse is wound to run 100 m. Whenever it encounters an obstacle (a pen boundary), it turns by uncertain angle. Try to estimate the lower boundary of the sum of those angles. -- Pertsel Vladimir <voldemar@wiscon.weizmann.ac.il>
Show solution
In each step except for the first and last, the mouse must turn from the boundary, make the step, then turn back to the boundary. The angle per unit length is minimum when the step is short. So the mouse hugs the boundary for most of the path. The total angle is now L - 4 sin(theta/2) + theta , where theta is the angle subtended by the chord which is the last(first) step. This is minimized by theta = 2PI/3 . The result is 100 - 2sqrt(3) + 2PI/3 ~ 98.63 -- Dave Ring <dwr2560@tam2000.tamu.edu>