In mathematics, the mice problem is a continuous

pursuit–evasion Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early ...

problem in which a number of mice (or insects, dogs, missiles, etc.) are considered to be placed at the corners of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

. In the classic setup, each then begins to move towards its immediate neighbour (clockwise or anticlockwise). The goal is often to find out at what time the mice meet. The most common version has the mice starting at the corners of a unit square, moving at unit speed. In this case they meet after a time of one unit, because the distance between two neighboring mice always decreases at a speed of one unit. More generally, for a regular polygon of

n

unit-length sides, the distance between neighboring mice decreases at a speed of

1 - \cos(2\pi/n)

, so they meet after a time of

1/\bigl(1 - \cos(2\pi/n)\bigr)

Path of the mice

For all regular polygons, each mouse traces out a

pursuit curve In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterized ...

in the shape of a

logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...

. These curves meet in the center of the polygon.

In media

In '' Dara Ó Briain: School of Hard Sums'', the mice problem is discussed. Instead of 4 mice, 4 ballroom dancers are used.

References

External links

*
Zeno's Mice (Ants) Problem and the Logarithmic Spirals
- YouTube lecture with equation derivation Recreational mathematics Pursuit–evasion {{geometry-stub