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# The Moving Sofa Problem

The main mathematical news

1968: John Hammersley proved that the upper bound of the area of a sofa that can be moved around an L-shape corner of width 1 unit is 2Ö2=~2.2074 square units. He designed such a sofa consisting of two quarter-circles with 1 1-unit radius connected by a rectangle of length 4/π.

1992: Joseph Gerver slightly improved Hammersley’s solution. The area of Gerver’s “sofa” is ~2.2195 square units.

2016: Romik presents a complete solution to the problem of moving a sofa around two opposite corners.

2017: Dan Romik and Yoav Kallus found (using a computer) a lower upper bound to the sofa moving problem, which is 2.37 square units.

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Additional Theorems / conjectures / Open questions
• The moving sofa problem, namely the minimum area of the shape that can be moved around an L-shaped corner is still open.
• The length measure of the largest rod that can be moved around a corner of width 1 unit is i√8≈8 units.
• Any shape that is enclosed within a circle whose diameter is equal to the width of the passage can move around a corner with or without turning it.

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The main mathematical concepts / Principles

MSC2020#51M25 Length, area, and volume in real geometry

MSC2020#97G60 Plane trigonometry

MSC2020#26A24 Differentiation (real functions of one variable)

MSC2020#97G30 Area and volume (educational aspects)

MSC2020#97I40 Differential calculus (educational aspects)

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