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# Moving Sofa Problem

What is the sofa of greatest area which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that

 (1)

(OEIS A086118). Gerver (1992) found a sofa with larger area and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver's sofa is a complicated shape composed of 18 arcs. Its area can be given by defining the constants , , , and by solving

 (2) (3) (4) (5)

This gives

 (6) (7) (8) (9)

Now define

 (10)

where

 (11) (12) (13)

Finally, define the functions

 (14) (15) (16)

The area of the optimal sofa is then given by

 (17) (18)

(Finch 2003).

Piano Mover's Problem

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## References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994.Finch, S. R. "Moving Sofa Constant." §8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.Gerver, J. L. "On Moving a Sofa Around a Corner." Geometriae Dedicata 42, 267-283, 1992.Sloane, N. J. A. Sequence A086118 in "The On-Line Encyclopedia of Integer Sequences."Stewart, I. Another Fine Math You've Got Me Into.... New York: W. H. Freeman, 1992.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 104, 2004. http://www.mathematicaguidebooks.org/.

## Referenced on Wolfram|Alpha

Moving Sofa Problem

## Cite this as:

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