Mice Problem

$n$ mice start at the corners of a regular $n$-gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a Spiral, meet in the center of the Polygon, and travel a distance

$$d_n={1\over 1-\cos\left({2\pi\over n}\right)}.$$

The first few values for $n=2$, 3, ..., are

${\textstyle{1\over 2}}, {\textstyle{2\over 3}}, 1, {\textstyle{1\over 5}}(5+\sqrt{5}), 2, {1\over 1-\cos\left({2\pi\over 7}\right)},$
$2+\sqrt{2}, {1\over 1-\cos\left({2\pi\over 9}\right)}, 3+\sqrt{5}, \ldots,$

giving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, ....

See also Apollonius Pursuit Problem, Pursuit Curve, Spiral, Tractrix

References

Bernhart, A. ``Polygons of Pursuit.'' Scripta Math. 24, 23-50, 1959.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 201-204, 1979.