Sources page biographical material

6.AG. MOVING AROUND A CORNER
There are several versions of this. The simplest is moving a ladder or board around a corner -- here the problem is two-dimensional and the ladder is thin enough to be considered as a line. There are slight variations -- the corner can be at a T or + junction; the widths of the corridors may differ; the angle may not be a right angle; etc. If the object being moved is thicker -- e.g. a table -- then the problem gets harder. If one can use the third dimension, it gets even harder.
H. E. Licks. Op. cit. in 5.A, 1917. Art. 110, p. 89. Stick going into a circular shaft in the ceiling. Gets [h^2/3 + d^2/3)]^3/2 for maximum length, where h is the height of the room and d is the diameter of the shaft. "A simple way to solve a problem which has proved a stumbling block to many."

Abraham. 1933. Prob. 82 -- Another ladder, pp. 37 & 45 (23 & 117). Ladder to go from one street to another, of different widths.

E. H. Johnson, proposer; W. B. Carver, solver. Problem E436. AMM 47 (1940) 569 & 48 (1941) 271 273. Table going through a doorway. Obtains 6th order equation.

J. S. Madachy. Turning corners. RMM 5 (Oct 1961) 37, 6 (Dec 1961) 61 & 8 (Apr 1962) 56. In 5, he asks for the greatest length of board which can be moved around a corner, assuming both corridors have the same width, that the board is thick and that vertical movement is allowed. In 6, he gives a numerical answer for his original values and asserts the maximal length for planar movement, with corridors of width w and plank of thickness t, is 2 (w2   t). In vol. 8, he says no two solutions have been the same.

L. Moser, proposer; M. Goldberg and J. Sebastian, solvers. Problem 66 11 -- Moving furniture through a hallway. SIAM Review 8 (1966) 381 382 & 11 (1969) 75 78 & 12 (1970) 582 586. "What is the largest area region which can be moved through a "hallway" of width one (see Fig. 1)?" The figure shows that he wants to move around a rectangular corner joining two hallways of width one. Sebastian (1970) studies the problem for moving an arc.

J. M. Hammersley. On the enfeeblement of mathematical skills .... Bull. Inst. Math. Appl. 4 (1968) 66 85. Appendix IV -- Problems, pp. 83 85, prob. 8, p. 84. Two corridors of width 1 at a corner. Show the largest object one can move around it has area < 2 2 and that there is an object of area  π/2 + 2/π = 2.2074.

Partial solution by T. A. Westwell, ibid. 5 (1969) 80, with editorial comment thereon on pp. 80 81.

T. J. Fletcher. Easy ways of going round the bend. MG 57 (No. 399) (Feb 1973) 16 22. Gives five methods for the ladder problem with corridors of different widths.

Neal R. Wagner. The sofa problem. AMM 83 (1976) 188 189. "What is the region of largest area which can be moved around a right angled corner in a corridor of width one?" Survey.

R. K. Guy. Monthly research problems, 1969 77. AMM 84 (1977) 807 815. P. 811 reports improvements on the sofa problem.

J. S. Madachy & R. R. Rowe. Problem 242 -- Turning table. JRM 9 (1976 77) 219 221.

G. P. Henderson, proposer; M. Goldberg, solver; M. S. Klamkin, commentator. Problem 427. CM 5 (1979) 77 & 6 (1979) 31 32 & 49 50. Easily finds maximal area of a rectangle going around a corner.

Research news: Conway's sofa problem. Mathematics Review 1:4 (Mar 1991) 5-8 & 32. Reports on Joseph Gerver's almost complete resolution of the problem in 1990. Says Conway asked the problem in the 1960s and that L. Moser is the first to publish it. Says a group at a convexity conference in Copenhagen improved Hammersley's results to 2.2164. Gerver's analysis gives an object made up of 18 segments with area 2.2195. The analysis depends on some unproven general assumptions which seem reasonable and is certainly the unique optimum solution given those assumptions.

A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5. Does usual problem, getting a quartic. The finds the shortest ladder. [This turns out to be the same as the longest ladder one can get around a corner from corridors of widths w and h, so 6.AG is related to 6.L.]