# Mice problem

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{{short description|Mathematical problem}}
{{for|the Monty Python sketch|The Mouse Problem}}
thumb|upright|Four mice
{{multiple image
|width=150
|image1=Problema dei topi n=3 animazione.gif
|caption1=Three mice
|image2=Problema dei topi n=6 animazione.gif
|caption2=Six mice
}}
In [mathematics](/source/mathematics), the '''mice problem''' is a continuous [pursuit–evasion](/source/pursuit%E2%80%93evasion) problem in which a number of mice (or insects, dogs, missiles, etc.) are considered to be placed at the corners of a [regular polygon](/source/regular_polygon). 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 <math>n</math> unit-length sides, the distance between neighboring mice decreases at a speed of <math>1 - \cos(2\pi/n)</math>, so they meet after a time of <math>1/\bigl(1 - \cos(2\pi/n)\bigr)</math>.<ref>{{cite book|last1=Gamow|first1=George|author1-link=George Gamow|last2=Stern|first2=Marvin|pages=112–114|publisher=Viking Press|title=Puzzle Math|year=1958}}</ref><ref>{{cite journal|last=Lucas|first=Édouard|author-link=Édouard Lucas|journal=Nouv. Corresp. Math.|pages=175–176|title=Problem of the Three Dogs|volume=3|year=1877}}</ref>

==Path of the mice==

For all regular polygons, each mouse traces out a [pursuit curve](/source/pursuit_curve) in the shape of a [logarithmic spiral](/source/logarithmic_spiral). These curves meet in the center of the polygon.<ref>{{cite journal
 | last = Bernhart | first = Arthur
 | journal = Scripta Mathematica
 | mr = 104178
 | pages = 23–50
 | title = Polygons of pursuit
 | volume = 24
 | year = 1959}}</ref>

== In media ==
In ''[Dara Ó Briain: School of Hard Sums](/source/Dara_%C3%93_Briain%3A_School_of_Hard_Sums)'', the mice problem is discussed. Instead of 4 mice, 4 ballroom dancers are used.<ref>{{Cite episode|title=Downton Abacus: The Maths of Wealth|first4=Marcus|minutes=24|number=4|season=3|date=March 2014|network=Dave|last4=Brigstocke|last3=Watson|series=Dara Ó Briain: School of Hard Sums|first3=Mark|last2=du Sautoy|first2=Marcus|last1=Ó Briain|first1=Dara|series-link=Dara_%C3%93_Briain:_School_of_Hard_Sums|language=English}}</ref>

== Whirl ==
thumb|A whirl with alternating black and white squares
Connecting the locations of the mice at different intervals forms a whirl: a figure consisting of a sequence of nested polygons, each smaller and rotated relative to the previous.<ref>Weisstein, Eric W., [http://mathworld.wolfram.com/Whirl.html ''Whirl''], Wolfram Mathworld</ref><ref>Freund, J. E. Introduction to Probability. New York: Dover, 1993.</ref><ref>Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 75, 2002.</ref><ref>Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, p. 66, 1991.</ref><ref>Pappas, T. "Spider & Spirals." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 228, 1989.</ref><ref>Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 201-202, 1991.</ref>

==References==
{{Reflist}}

== External links ==
*{{mathworld|title=Mice Problem|id=MiceProblem}}
* [https://www.youtube.com/watch?v=NdTVvWrD6r0 Zeno's Mice (Ants) Problem and the Logarithmic Spirals] - YouTube lecture with equation derivation

Category:Recreational mathematics
Category:Pursuit–evasion

{{geometry-stub}}

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Adapted from the Wikipedia article [Mice problem](https://en.wikipedia.org/wiki/Mice_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Mice_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.