Sources page biographical material

6.J. FOUR BUGS AND OTHER PURSUIT PROBLEMS
The general problem becomes too technical to remain recreational, so I will not try to be exhaustive here.
Arthur Bernhart.

Curves of pursuit. SM 20 (1954) 125 141.

Curves of pursuit -- II. SM 23 (1957) 49 65.

Polygons of pursuit. SM 24 (1959) 23 50.

Curves of general pursuit. SM 24 (1959) 189 206.

Extensive history and analysis. First article covers one dimensional pursuit, then two dimensional linear pursuit. Second article deals with circular pursuit. Third article is the 'four bugs' problem -- analysis of equilateral triangle, square, scalene triangle, general polygon, Brocard points, etc. Last article includes such variants as variable speed, the tractrix, miscellaneous curves, etc.

Mr. Ash, proposer; editorial note saying there is no solver. Ladies' Diary, 1748-47 = T. Leybourn, II: 15-17, quest. 310, with 'Solution by ΦIΛΟΠΟΝΟΣ, taken from Turner's Exercises, where this question was afterwards proposed and answered ...' A fly is constrained to move on the periphery of a circle. Spider starts 30^o away from the fly, but walks across the circle, always aiming at the fly. If she catches the fly 180^o from her starting point, find the ratio of their speeds. ΦIΛΟΠΟΝΟΣ solves the more general problem of finding the curve when the spider starts anywhere.

Carlile. Collection. 1793. Prob. CV, p. 62. A dog and a duck are in a circular pond of radius 40 and they swim at the same speed. The duck is at the edge and swims around the circumference. The dog starts at the centre and always swims toward the duck, so the dog and the duck are always on a radius. How far does the dog swim in catching the duck? He simply gives the result as 20π. Letting R be the radius of the pond and V be the common speed, I find the radius of the dog, r, is given by r = R sin Vt/R. Since the angle, θ, of both the duck and the dog is given by θ = Vt/R, the polar equation of the dog's path is r = R sin θ and the path is a semicircle whose diameter is the appropriate radius perpendicular to the radius to the duck's initial position.

Cambridge Math. Tripos examination, 5 Jan 1871, 9 to 12. Problem 16, set by R. K. Miller. Three bugs in general position, but with velocities adjusted to make paths similar and keep the triangle similar to the original.

Lucas. (Problem of three dogs.) Nouvelle Correspondance Mathématique 3 (1877) 175 176. ??NYS -- English in Arc., AMM 28 (1921) 184 185 & Bernhart.

H. Brocard. (Solution of Lucas' problem.) Nouv. Corr. Math. 3 (1877) 280. ??NYS -- English in Bernhart.

Pearson. 1907. Part II, no. 66: A duck hunt, pp. 66 & 172. Duck swims around edge of pond; spaniel starts for it from the centre at the same speed.

A. S. Hathaway, proposer and solver. Problem 2801. AMM 27 (1920) 31 & 28 (1921) 93 97. Pursuit of a prey moving on a circle. Morley's and other solutions fail to deal with the case when the velocities are equal. Hathaway resolves this and shows the prey is then not caught.

F. V. Morley. A curve of pursuit. AMM 28 (1921) 54-61. Graphical solution of Hathaway's problem.

R. C. Archibald [Arc.] & H. P. Manning. Remarks and historical notes on problems 19 [1894], 160 [1902], 273 [1909] & 2801 [1920]. AMM 28 (1921) 91-93.

W. W. Rouse Ball. Problems -- Notes: 17: Curves of pursuit. AMM 28 (1921) 278 279.

A. H. Wilson. Note 19: A curve of pursuit. AMM 28 (1921) 327.

Editor's note to Prob. 2 (proposed by T. A. Bickerstaff), National Mathematics Magazine (1937/38) 417 cites Morley and Archibald and adds that some authors credit the problem to Leonardo da Vinci -- e.g. MG (1930-31) 436 -- ??NYS

Nelson F. Beeler & Franklyn M. Branley. Experiments in Optical Illusion. Ill. by Fred H. Lyon. Crowell, 1951, An illusion doodle, pp. 68-71, describes the pattern formed by four bugs starting at the corners of a square, drawing the lines of sight at (approximately) regular intervals. Putting several of the squares together, usually with alternating directions of motion, gives a pleasant pattern which is now fairly common. They call this 'Huddy's Doodle', but give no source.

J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. 'Lion and man', pp. 135 136 (114 117). The 1986 ed. adds three diagrams and revises the text somewhat. I quote from it. "A lion and a man in a closed circular arena have equal maximum speeds. What tactics should the lion employ to be sure of his meal?" This was "invented by R. Rado in the late thirties" and "swept the country 25 years later". [The 1953 ed., says Rado didn't publish it.] The correct solution "was discovered by Professor A. S. Besicovitch in 1952". [The 1953 ed. says "This has just been discovered ...; here is the first (and only) version in print."]

C. C. Puckette. The curve of pursuit. MG 37 (No. 322) (Dec 1953) 256 260. Gives the history from Bouguer in 1732. Solves a variant of the problem.

R. H. Macmillan. Curves of pursuit. MG 40 (No. 331) (Feb 1956) 1 4. Fighter pursuing bomber flying in a straight line. Discusses firing lead and acceleration problems.

Gamow & Stern. 1958. Homing missiles. Pp. 112 114.

Howard D. Grossman, proposer; unspecified solver. Problem 66 -- The walk around. In: L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 40 & 203 205. Four bugs -- asserts Grossman originated the problem.

I. J. Good. Pursuit curves and mathematical art. MG 43 (No. 343) (Feb 1959) 34 35. Draws tangent to the pursuit curves in an equilateral triangle and constructs various patterns with them. Says a similar but much simpler pattern was given by G. B. Robison; Doodles; AMM 61 (1954) 381-386, but Robison's doodles are not related to pursuit curves, though they may have inspired Good to use the pursuit curves.

J. Charles Clapham. Playful mice. RMM 10 (Aug 1962) 6 7. Easy derivation of the distance travelled for n bugs at corners of a regular n gon. [I don't see this result in Bernhart.]

C. G. Paradine. Note 3108: Pursuit curves. MG 48 (No. 366) (Dec 1964) 437 439. Says Good makes an error in Note 3079. He shows the length of the pursuit curve in the equilateral triangle is ⅔ of the side and describes the curve as an equiangular spiral. Gives a simple proof that the length of the pursuit curve in the regular n gon is the side divided by (1   cos 2π/n).

M. S. Klamkin & D. J. Newman. Cyclic pursuit or "The three bugs problem". AMM 78 (1971) 631 639. General treatment. Cites Bernhart's four SM papers and some of the history therein.

P. K. Arvind. A symmetrical pursuit problem on the sphere and the hyperbolic plane. MG 78 (No. 481) (Mar 1994) 30-36. Treats the n bugs problems on the surfaces named.

Barry Lewis. A mathematical pursuit. M500 170 (Oct 1999) 1-8. Starts with equilateral triangular case, giving QBASIC programs to draw the curves as well as explicit solutions. Then considers regular n-gons. Then considers simple pursuit, one beast pursuing another while the other moves along some given path. Considers the path as a straight line or a circle. For the circle, he asserts that the analytic solution was not determined until 1926, but gives no reference.