Sources page biographical material

6.G. SOMA CUBE
Piet Hein invented the Soma Cube in 1936. (S&B, pp. 40 41.) ??Is there any patent??

M. Gardner. SA (Sep 1958) = 2nd Book, Chap 6.

Richard K. Guy. Loc. cit. in 5.H.2, 1960. Pp. 150-151 discusses cubical solutions -- 234 found so far. He proposes the 'bath' shape -- a 5 x 3 x 2 cuboid with a 3 x 1 x 1 hole in the top layer. In a 1985 letter, he said that O'Beirne had introduced the Soma to him and his family. in 1959 and they found 234 solutions before Mike Guy went to Cambridge -- see below.

P. Hein, et al. Soma booklet. Parker Bros., 1969, 56pp. Asserts there are 240 simple solutions and 1,105,920 total solutions, found by J. H. Conway & M. J. T. Guy with a a computer (but cf Gardner, below) and by several others. [There seem to be several versions of this booklet, of various sizes.]

Thomas V. Atwater, ed. Soma Addict. 4 issues, 1970 1971, produced by Parker Brothers. (Gardner, below, says only three issues appeared.) ??NYS -- can anyone provide a set or photocopies??

M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. States there are 240 solutions for the cube, obtained by many programs, but first found by J. H. Conway & M. J. T. Guy in 1962, who did not use a computer, but did it by hand "one wet afternoon". Richard Guy's 1985 letter notes that Mike Guy had a copy of the Guy family's 234 solutions with him.

SOMAP ??NYS -- ??details. (Schaaf III 52)

Winning Ways, 1982, II, 802 803 gives the SOMAP.

Jon Brunvall et al. The computerized Soma Cube. Comp. & Maths. with Appl. 12B:1/2 (1986 [Special issues 1/2 & 3/4 were separately printed as: I. Hargittai, ed.; Symmetry -- Unifying Human Understanding; Pergamon, 1986.] 113 121. They cite Gardner's 2nd Book which says the number of solutions is unknown and they use a computer to find them.
6.G.1. OTHER CUBE DISSECTIONS
See also 6.N, 6.U.2, 6.AY.1 and 6.BJ. The predecessors of these puzzles seem to be the binomial and trinomial cubes showing (a+b)³ and (a+b+c)³. I have an example of the latter from the late 19C. Here I will consider only cuts parallel to the cube faces -- cubes with cuts at angles to the faces are in 6.BJ. Most of the problems here involve several types of piece -- see 6.U.2 for packing with one kind of piece.
Catel. Kunst-Cabinet. 1790. Der algebraische Würfel, p. 6 & fig 50 on plate II. Shows a binomial cube: (a + b)³ = a³ + 3a²b + 3ab² + b³.

Bestelmeier. 1801. Item 309 is a binomial cube, as in Catel. "Ein zerschnittener Würfel, mit welchem die Entstehung eines Cubus, dessen Seiten in 2 ungleiche Theile a + b getheilet ist, gezeigt ist."

Hoffmann. 1893. Chap. III, no. 39: The diabolical cube, pp. 108 & 142 = Hoffmann-Hordern, pp. 108-109, with photos. 6: 0, 1, 1, 1, 1, 1, 1, i.e. six pieces of volumes 2, 3, 4, 5, 6, 7. Photos on p. 108 shows Cube Diabolique and its box, by Watilliaux, dated 1874-1895.

J. G. Mikusiński. French patent. ??NYS -- cited by Steinhaus.

H. Steinhaus. Mikusiński's Cube. Mathematical Snapshots. Not in Stechert, 1938, ed. OUP, NY: 1950: pp. 140 142 & 263; 1960, pp. 179 181 & 326; 1969 (1983): pp. 168-169 & 303.

John Conway. In an email of 7 Apr 2000, he says he developed the dissection of the 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2 in c1960 and then adapted it to the 5 x 5 x 5 into 3  1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13 1 x 2 x 4 and the 5 x 5 x 5 into 3  1 x 1 x 3 and 29 1 x 2 x 2. He says his first publication of it was in Winning Ways, 1982 (cf below).

Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2. [Jan de Geus has sent a photocopy of some of this but it does not cover this topic.]

M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. Discusses Hoffmann's Diabolical Cube and Mikusiński's cube. Says he has 8 solutions for the first and that there are just 2 for the second. The Addendum reports that Wade E. Philpott showed there are just 13 solutions of the Diabolical Cube. Conway has confirmed this. Gardner briefly describes the solutions. Gardner also shows the Lesk Cube, designed by Lesk Kokay (Mathematical Digest [New Zealand] 58 (1978) ??NYS), which has at least 3 solutions.

D. A. Klarner. Brick packing puzzles. JRM 6 (1973) 112 117. Discusses 3 x 3 x 3 into 3  1 x 1 x 1 and 6 1 x 2 x 2 attributed to Slothouber Graatsma; Conway's 5 x 5 x 5 into 3  1 x 1 x 3 and 29 1 x 2 x 2; Conway's 5 x 5 x 5 into 3 1 x 1 x 3, 1 2 x 2 x 2, 1  1 x 2 x 2 and 13 1 x 2 x 4. Because of the attribution to Slothouber & Graatsma and not knowing the date of Conway's work, I had generally attributed the 3 x 3 x 3 puzzle to them and Stewart Coffin followed this in his book. However, it now seems that it really is Conway's invention and I must apologize for misleading people.

Leisure Dynamics, the US distributor of Impuzzables, a series of 6 3 x 3 x 3 cube dissections identified by colours, writes that they were invented by Robert Beck, Custom Concepts Inc., Minneapolis. However, the Addendum to Gardner, above, says they were designed by Gerard D'Arcey.

Winning Ways. 1982. Vol. 2, pp. 736-737 & 801. Gives the 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2 and the 5 x 5 x 5 into 3  1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13  1 x 2 x 4, which is called Blocks-in-a-Box. No mention of the other 5 x 5 x 5. Mentions Foregger & Mather, cf in 6.U.2.

Michael Keller. Polycube update. World Game Review 4 (Feb 1985) 13. Reports results of computer searches for solutions. Hoffmann's Diabolical Cube has 13; Mikusinski's Cube has 2; Soma Cube has 240; Impuzzables: White -- 1; Red -- 1; Green -- 16; Blue -- 8; Orange -- 30; Yellow -- 1142.

Michael Keller. Polyform update. World Game Review 7 (Oct 1987) 10 13. Says that Nob Yoshigahara has solved a problem posed by O'Beirne: How many ways can 9  L trominoes make a cube? Answer is 111. Gardner, Knotted, chap. 3, mentioned this. Says there are solutions with n L trominoes and 9 n straight trominoes for n  1 and there are 4 solutions for n = 0. Says the Lesk Cube has 4 solutions. Says Naef's Gemini Puzzle was designed by Toshiaki Betsumiya. It consists of the 10 ways to join two 1 x 2 x 2 blocks.

H. J. M. van Grol. Rik's Cube Kit -- Solid Block Puzzles. Analysis of all 3 x 3 x 3 unit solid block puzzles with non planar 4 unit and 5 unit shapes. Published by the author, The Hague, 1989, 16pp. There are 3 non planar tetracubes and 17 non planar pentacubes. A 3 x 3 x 3 cube will require the 3 non planar tetracubes and 3 of the non planar pentacubes -- assuming no repeated pieces. He finds 190 subsets which can form cubes, in 1 to 10 different ways.

Nob Yoshigahara. (Title in Japanese: (Puzzle in Wood)). H. Tokuda, Sowa Shuppan, Japan, 1987. Pp. 68-69 is a 3^3 designed by Nob -- 6: 01005.
6.G.2. DISSECTION OF 6³ INTO 3³, 4³ AND 5³, ETC.
H. W. Richmond. Note 1672: A geometrical problem. MG 27 (No. 275) (Jul 1943) 142. AND Note 1704: Solution of a geometrical problem (Note 1672). MG 28 (No. 278) (Feb 1944) 31 32. Poses the problem of making such a dissection, then gives a solution in 12 pieces: three 1 x 3 x 3; 4 x 4 x 4; four 1 x 5 x 5; 1 x 4 x 4; two 1 x 1 x 2 and a V pentacube.

Anon. [= John Leech, according to Gardner, below]. Two dissection problems, no. 2. Eureka 13 (Oct 1950) 6 & 14 (Oct 1951) 23. Asks for such a dissection using at most 10 pieces. Gives an 8 piece solution due to R. F. Wheeler. [Cundy & Rollett; Mathematical Models; 2nd ed., pp. 203 205, say Eureka is the first appearance they know of this problem. See Gardner, below, for the identity of Leech.]

Richard K. Guy. Loc. cit. in 5.H.2, 1960. Mentions the 8 piece solution.

J. H. Cadwell. Some dissection problems involving sums of cubes. MG 48 (No. 366) (Dec 1964) 391 396. Notes an error in Cundy & Rollett's account of the Eureka problem. Finds examples for 12³ + 1³ = 10³ + 9³ with 9 pieces and 9³ = 8³ + 6³ + 1³ with 9 pieces.

J. H. Cadwell. Note 3278: A three way dissection based on Ramanujan's number. MG 54 (No. 390) (Dec 1970) 385 387. 7 x 13 x 19 to 10³ + 9³ and 12³ + 1³ using 12 pieces.

M. Gardner. SA (Oct 1973) c= Knotted, chap. 16. He says that the problem was posed by John Leech. He gives Wheeler's initials as E. H. ?? He says that J. H. Thewlis found a simpler 8 piece solution, further simplified by T. H. O'Beirne, which keeps the 4 x 4 x 4 cube intact. This is shown in Gardner. Gardner also shows an 8 piece solution which keeps the 5 x 5 x 5 intact, due to E. J. Duffy, 1970. O'Beirne showed that an 8 piece dissection into blocks is impossible and found a 9 block solution in 1971, also shown in Gardner.

Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1984. Section 24.1, pp. 118 120 gives Wheeler's solution and admires it.

Richard K. Guy, proposer; editors & Charles H. Jepson [should be Jepsen], partial solvers. Problem 1122. CM 12 (1987) 50 & 13 (1987) 197 198. Asks for such dissections under various conditions, of which (b) is the form given in Eureka. Eight pieces is minimal in one case and seems minimal in two other cases. Eleven pieces is best known for the first case, where the pieces must be blocks, but this appears to be the problem solved by O'Beirne in 1971, reported in Gardner, above.

Charles H. Jepsen. Additional comment on Problem 1122. CM 14 (1988) 204 206. Gives a ten piece solution of the first case.

Chris Pile. Cube dissection. M500 134 (Aug 1993) 2-3. He feels the 1 x 1 x 2 piece occurring in Cundy & Rollett is too small and he provides another solution with 8 pieces, the smallest of which contains 8 unit cubes. Asks how uniform the piece sizes can be.

6.G.3. DISSECTION OF A DIE INTO NINE 1 x 1 x 3
Hoffmann. 1893. Chap. III, no. 17: The "Spots" puzzle, pp. 98 99 & 130 131 = Hoffmann Hordern, pp. 90-91, with photo. Says it is made by Wolff & Son. Photo on p. 91 shows an example made by E. Wolff & Son, London.

Benson. 1904. The spots puzzle, pp. 203 204. As in Hoffmann.

Collins. Book of Puzzles. 1927. Pp. 131 134: The dissected die puzzle. The solution is different than Hoffmann's.

Rohrbough. Puzzle Craft. 1932. P. 21 shows a dissected die, but with no text. The picture is the same as in Hoffmann's solution.

Slocum. Compendium. Shows Diabolical Dice from Johnson Smith catalogue, 1935.

Harold Cataquet. The Spots puzzle revisited. CFF 33 (Feb 1994) 20-21. Brief discussion of two versions.

David Singmaster. Comment on the "Spots" puzzle. 29 Sep 1994, 2pp. Letter in response to the above. I note that there is no standard pattern for a die other than the opposite sides adding to seven. There are 2³ = 8 ways to orient the spots forming 2, 3, and 6. There are two handednesses, so there are 16 dice altogether. (This was pointed out to me perhaps 10 years before by Richard Guy and Ray Bathke. I have since collected examples of all 16 dice.) However, Ray Bathke showed me Oriental dice with the two spots of the 2 placed horizontal or vertically rather than diagonally, giving another 16 dice (I have 5 types), making 32 dice in all. A die can be dissected into 9 1 x 1 x 3 pieces in 6 ways if the layers have to alternate in direction, or in 21 ways in general. I then pose a number of questions about such dissections.
6.G.4. USE OF OTHER POLYHEDRAL PIECES
S&B. 1986. P. 42 shows Stewart Coffin's 'Pyramid Puzzle' using pieces made from truncated octahedra and his 'Setting Hen' using pieces made from rhombic dodecahedra. Coffin probably devised these in the 1960s -- perhaps his book has some details of the origins of these ideas. ??check.

Mark Owen & Matthew Richards. A song of six splats. Eureka 47 (1987) 53 58. There are six ways to join three truncated octahedra. For reasons unknown, these are called 3 splats. They give various shapes which can and which cannot be constructed from the six 3 splats.

6.H. PICK'S THEOREM
Georg Pick. Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen naturwissenschaftlich medicinischen Vereines für Böhmen "Lotos" in Prag (NS) 19 (1899) 311 319. Pp. 311 314 gives the proof, for an oblique lattice. Pp. 318 319 gives the extension to multiply connected and separated regions. Rest relates to number theory. [I have made a translation of the material on Pick's Theorem.]

Charles Howard Hinton. The Fourth Dimension. Swan Sonnenschein & Co., London, 1906. Metageometry, pp. 46-60. [This material is in Speculations on the Fourth Dimension, ed. by R. v. B. Rucker; Dover, 1980, pp. 130-141. Rucker says the book was published in 1904, so my copy may be a reprint??] In the beginning of this section, he draws quadrilateral shapes on the square lattice and determines the area by counting points, but he counts I + E/2 + C/4, which works for quadrilaterals but is not valid in general.

H. Steinhaus. O mierzeniu pól płaskich. Przegląd Matematyczno Fizyczny 2 (1924) 24 29. Gives a version of Pick's theorem, but doesn't cite Pick. (My thanks to A. Mąkowski for an English summary of this.)

H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938, pp. 16-17 & 132. OUP, NY: 1950: pp. 76 77 & 260 (note 77); 1960: pp. 99 100 & 324 (note 95); 1969 (1983): pp. 96 97 & 301 (note 107). In 1938 he simply notes the theorem and gives one example. In 1950, he outlines Pick's argument. He refers to Pick's paper, but in "Ztschr. d. Vereins 'Lotos' in Prag". Steinhaus also cites his own paper, above.

J. F. Reeve. On the volume of lattice polyhedra. Proc. London Math. Soc. 7 (1957) 378 395. Deals with the failure of the obvious form of Pick's theorem in 3 D and finds a valid generalization.

Ivan Niven & H. S. Zuckerman. Lattice points and polygonal area. AMM 74 (1967) 1195 1200. Straightforward proof. Mention failure for tetrahedra.

D. W. De Temple & J. M. Robertson. The equivalence of Euler's and Pick's theorems. MTr 67 (1974) 222 226. ??NYS.

W. W. Funkenbusch. From Euler's formula to Pick's formula using an edge theorem. AMM 81 (1974) 647 648. Easy proof though it could be easier.

R. W. Gaskell, M. S. Klamkin & P. Watson. Triangulations and Pick's theorem. MM 49 (1976) 35 37. A bit roundabout.

Richard A. Gibbs. Pick iff Euler. MM 49 (1976) 158. Cites DeTemple & Robertson and observes that both Pick and Euler can be proven from a result on triangulations.

John Reay. Areas of hex-lattice polygons, with short sides. Abstracts Amer. Math. Soc. 8:2 (1987) 174, #832-51-55. Gives a formula for the area in terms of the boundary and interior points and the characteristic of the boundary, but it is an open question to determine when this formula gives the actual area.
6.I. SYLVESTER'S PROBLEM OF COLLINEAR POINTS
If a set of non collinear points in the plane is such that the line through any two points of the set contains a third point of the set, then the set is infinite.
J. J. Sylvester. Question 11851. The Educational Times 46 (NS, No. 383) (1 Mar 1893) 156.

H. J. Woodall & editorial comment. Solution to Question 11851. Ibid. (No. 385) (1 May 1893) 231. A very spurious solution.

(The above two items appear together in Math. Quest. with their Sol. Educ. Times 59 (1893) 98 99.)

E. Melchior. Über Vielseite der projecktiven Ebene. Deutsche Math. 5 (1940) 461 475. Solution, but in a dual form.

P. Erdös, proposer; R. Steinberg, solver & editorial comment giving solution of T. Grünwald (later = T. Gallai). Problem 4065. AMM 50 (1943) 65 & 51 (1944) 169 171.

L. M. Kelly. (Solution.) In: H. S. M. Coxeter; A problem of collinear points; AMM 55 (1948) 26 28. Kelly's solution is on p. 28.

G. A. Dirac. Note 2271: On a property of circles. MG 36 (No. 315) (Feb 1952) 53 54. Replace 'line' by 'circle' in the problem. He shows this is true by inversion. He asks for an independent proof of the result, even for the case when two, three are replaced by three, four.

D. W. Lang. Note 2577: The dual of a well known theorem. MG 39 (No. 330) (Dec 1955) 314. Proves the dual easily.

H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 4.7: Sylvester's problem of collinear points, pp. 65-66. Sketches history and gives Kelly's proof.

W. O. J. Moser. Sylvester's problem, generalizations and relatives. In his: Research Problems in Discrete Geometry 1981, McGill University, Montreal, 1981. Section 27, pp. 27 1 -- 27 14. Survey with 73 references. (This problem is not in Part 1 of the 1984 ed. nor in the 1986 ed.)

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.

6.K. DUDENEY'S SQUARE TO TRIANGLE DISSECTION
Dudeney. Weekly Dispatch (6 Apr, 20 Apr, 4 May, 1902) all p. 13.

Dudeney. The haberdasher's puzzle. London Mag. 11 (No. 64) (Nov 1903) 441 & 443. (Issue with solution not found.)

Dudeney. Daily Mail (1 & 8 Feb 1905) both p. 7.

Dudeney. CP. 1907. Prob. 25: The haberdasher's puzzle, pp. 49 50 & 178 180.

Western Puzzle Works, 1926 Catalogue. No. 1712 -- unnamed, but shows both the square and the triangle. Apparently a four piece puzzle.

M. Adams. Puzzle Book. 1939. Prob. C.153: Squaring a triangle, pp. 162 & 189. Asserts that Dudeney's method works for any triangle, but his example is close to equilateral and I recall that this has been studied and only certain shapes will work??

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 40-41. Extends to dissecting a quadrilateral to a specified triangle and gives a number of related problems.