The Moving Sofa Problem (in Turkish: Taşınan Kanepe Problemi) is a geometric optimization problem formulated in two-dimensional Euclidean space. Its exact solution remains unknown to this day. The problem seeks to determine the two-dimensional shape—resembling a sofa—that has the largest possible area and can be maneuvered around a right-angled hallway corner. In this context, the “sofa” refers to any two-dimensional shape that can be both rotated and translated.

The problem was first formulated in 1966 by mathematician Leo Moser. Since then, it has attracted interest from both theoretical mathematicians and researchers working in applied geometry.

Mathematical Framework

The problem is typically stated as follows:

What is the two-dimensional shape with the maximum possible area that can be moved (using translations and rotations) around a 90° corner formed by two hallways of unit width?

The objective is to determine the shape that maximizes area while still being able to pass through the corner. The challenge lies not in the shape’s exact dimensions but in maximizing the area under the constraint of successful passage. The complexity of the problem stems from the large number of feasible shapes and the geometric constraints involving rotation, collision, and boundary conditions.

Known Results

A definitive solution has not yet been found. However, researchers have established various lower and upper bounds:

Lower Bound: In 1992, Joseph Gerver proposed a complex shape with an area of approximately 2.2195 square units. Gerver’s shape is composed of 18 parabolic arcs and is currently the largest known valid solution to the moving sofa problem.

^{The Sofa Moving Problem – Gerver’s Proposed Solution Analogy (Generated by Artificial Intelligence)}

Upper Bound: In 2018, Dan Romik and his collaborators established an upper bound for the moving sofa problem using numerical methods. They determined that the maximum area cannot exceed approximately 2.37 square units. This result implies that no valid sofa shape—one that can successfully pass through the corner—can have an area larger than this value.

^{The Sofa Moving Problem – Romik’s Proposed Solution Analogy (Generated by Artificial Intelligence)}

Derivative Problems

A well-known variant of the moving sofa problem is called the “sofa mover’s problem.” In this version, the goal is not only to determine the optimal shape but also to optimize the path the shape takes while moving through the corner. Other more complex versions have also been explored, including cases involving non-right-angled corridors, narrow passages, or multiple turns.

Applications and Mathematical Significance

The moving sofa problem is not confined to theoretical interest alone. It has practical analogies in fields such as robotics, motion planning, navigation in tight spaces, and AI-assisted transport simulations. Moreover, it is notable for combining concepts from various mathematical subfields, including geometry, analysis, and computational optimization, making it a rich subject for interdisciplinary research.