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6.M. SPIDER & FLY PROBLEMS
These involve finding the shortest distance over the surface of a cube or cylinder. I've just added the cylindrical form -- see Dudeney (1926), Perelman and Singmaster. The shortest route from a corner of a cube or cuboid to a diagonally opposite corner must date back several centuries, but I haven't seen any version before 1937! I don't know if other shapes have been done -- the regular (and other) polyhedra and the cone could be considered.

Two-dimensional problems are in 10.U.

Loyd. The Inquirer (May 1900). Gives the Cyclopedia problem. ??NYS -- stated in a letter from Shortz.

Dudeney. Problem 501 -- The spider and the fly. Weekly Dispatch (14 & 28 Jun 1903) both p. 16. 4 side version.

Dudeney. Breakfast table problems, No. 320 -- The spider and the fly. Daily Mail (18 & 21 Jan 1905) both p. 7. Same as the above problem.

Dudeney. Master of the breakfast table problem. Daily Mail (1 & 8 Feb 1905) both p. 7. Interview with Dudeney in which he gives the 5 side version.

Ball. MRE, 4th ed., 1905, p. 66. Gives the 5 side version, citing the Daily Mail of 1 Feb 1905. He says he heard a similar problem in 1903 -- presumably Dudeney's first version. In the 5th ed., 1911, p. 73, he attributes the problem to Dudeney.

Dudeney. CP. 1907. Prob. 75: The spider and the fly, pp. 121 122 & 221 222. 5 side version with discussion of various generalizations.

Dudeney. The world's best problems. 1908. Op. cit. in 2. P. 786 gives the five side version.

Sidney J. Miller. Some novel picture puzzles -- No. 6. Strand Mag. 41 (No. 243) (Mar 1911) 372 & 41 (No. 244) (Apr 1911) 506. Contest between two snails. Better method uses four sides, similar to Dudeney's version, but with different numbers.

Loyd. The electrical problem. Cyclopedia, 1914, pp. 219 & 368 (= MPSL2, prob. 149, pp. 106 & 169 = SLAHP: Wiring the hall, pp. 72 & 114). Same as Dudeney's first, four side, version. (In MPSL2, Gardner says Loyd has simplified Dudeney's 5 side problem. More likely(?) Loyd had only seen Dudeney's earlier 4 side problem.)

Dudeney. MP. 1926. Prob. 162: The fly and the honey, pp. 67 & 157. (= 536, prob. 325, pp. 112 & 313.) Cylindrical problem.

Perelman. FFF. 1934. The way of the fly. 1957: Prob. 68, pp. 111 112 & 117 118; 1979: Prob. 72, pp. 136 & 142 144. MCBF: Prob. 72, pp. 134 & 141-142. Cylindrical form, but with different numbers and arrangement than Dudeney's MP problem.

Haldeman-Julius. 1937. No. 34: The louse problem, pp. 6 & 22. Room 40 x 20 x 10 with louse at a corner wanting to go to a diagonally opposite corner. Problem sent in by J. R. Reed of Emmett, Idaho. Answer is 50!

M. Kraitchik. Mathematical Recreations, 1943, op. cit. in 4.A.2, chap. 1, prob. 7, pp. 17 21. Room with 8 equal routes from spider to fly. (Not in his Math. des Jeux.)

Sullivan. Unusual. 1943. Prob. 10: Why not fly? Find shortest route from a corner of a cube to the diagonally opposite corner.

William R. Ransom. One Hundred Mathematical Curiosities. J. Weston Walch, Portland, Maine, 1955. The spider problem, pp. 144 146. There are three types of path, covering 3, 4 and 5 sides. He determines their relative sizes as functions of the room dimensions.

Birtwistle. Math. Puzzles & Perplexities. 1971.

Round the cone, pp. 144 & 195. What is the shortest distance from a point P around a cone and back to P? Answer is "An ellipse", which doesn't seem to answer the question. If the cone has height H, radius R and P is l from the apex, then the slant height L is (R² + H²), the angle of the opened out cone is θ = 2πR/L and the required distance is 2l sin θ/2.

Spider circuit, pp. 144 & 198. Spider is at the midpoint of an edge of a cube. He wants to walk on each of the faces and return. What is his shortest route? Answer is "A regular hexagon. (This may be demonstrated by putting a rubber band around a cube.)"

David Singmaster. The spider spied her. Problem used as: More than one way to catch a fly, The Weekend Telegraph (2 Apr 1984). Spider inside a glass tube, open at both ends, goes directly toward a fly on the outside. When are there two equally short paths? Can there be more than two shortest routes?

Yoshiyuki Kotani has posed the following general and difficult problem. On an a x b x c cuboid, which two points are furthest apart, as measured by an ant on the surface? Dick Hess has done some work on this, but I believe that even the case of square cross-section is not fully resolved.