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Mice Problem

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In the mice problem, also called the beetle 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 logarithmic spiral, meet in the center of the polygon, and travel a distance

d_n=1/(1-cos((2pi)/n)).

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

1/2,2/3,1,1/5(5+sqrt(5)),2,1/(1-cos((2pi)/7)), 2+sqrt(2),1/(1-cos((2pi)/9)),3+sqrt(5),...,

giving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.

The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original.

See also

Apollonius Pursuit Problem, Brocard Points, Pursuit Curve, Spiral, Tractrix, Whirl

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References

Bernhart, A. "Polygons of Pursuit." Scripta Math. 24, 23-50, 1959.Brocard, H. "Solution of Lucas's Problem." Nouv. Corresp. Math. 3, 280, 1877.Clapham, A. J. Rec. Math. Mag., Aug. 1962.Gardner, M. The Scientific American Book of Mathematical Puzzles and Diversions. New York: NY: Simon and Schuster, 1959.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 240-243, 1984.Good, I. J. "Pursuit Curves and Mathematical Art." Math. Gaz. 43, 34-35, 1959.Lucas, E. "Problem of the Three Dogs." Nouv. Corresp. Math. 3, 175-176, 1877.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 201-204, 1979.Marshall, J. A.; Broucke, M. E.; and Francis, B. A. "Pursuit Formations of Unicycles." Automata 41, 2005. http://www.control.toronto.edu/~marshall/docs/MarBroFra-Auto-4141-final.pdf.Miller, R. K. Problem 16. Cambridge Math. Tripos Exam. January 5, 1871.Nester, D. "Mathematics Seminar: Beetle Centers of Triangles." http://www.bluffton.edu/mat/dept/seminar_docs/BeetleCenters/.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 136, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 201-202, 1991.Wilson, J. "Problem: Four Dogs." http://jwilson.coe.uga.edu/emt725/Four.Dogs/four.dogs.html.

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Mice Problem

Cite this as:

Weisstein, Eric W. "Mice Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MiceProblem.html

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