Interestingly, this is one of the few problems in geometry that do not have any defined solution. It was invented in 1999 by a phenomenal mathematician and many variations of the problem have emerged since then. Do you think you have what it takes to solve this problem? Take this quiz to ready yourself first.
Adam Logan
Newt Moser
Leo Moser
Jones Moser
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1965
1966
1967
1968
1969
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Panraksa, Wetzel, Wichiramala
Gerriets, Poole, Wetzel
Leo Moser, Wichiramala, Panraksa
Blaschke, Panraksa, Wetzel
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Pythagoras
Avogadro
Blaschke
Venn
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The problem asks for the region of smallest area that can accommodate every plane curve of length.
It asks whether every problem whose solution can be quickly verified (technically, verified in polynomial time) can also be solved quickly (again, in polynomial time).
Providing grids, group theory that is used to describe ideas of when two grids are equivalent, and computational complexity with regards to solving problems.
A mathematical problem that involves a hexagonal lattice.
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Logic
Mathematical physics
Calculus and analysis
Geometry and topology
Dynamical systems and differential equations
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It means that the curve may be rotated and translated to fit inside the region.
The existence of a solution follows from the Blaschke selection theorem.
It is not trivial to determine whether a given shape forms a solution.
It is not completely trivial that a solution exists.
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This means that any solution region must have area at least (0.43893)/2 _> 0.21946.
The area of a solution region to the problem lies between 0.21946 and 0.27524.
The problem asks for the region of smallest area that can accommodate every plane curve of length 1.
It is easy to show that the circular disk with diameter 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539.
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Rectangles
Squares
Rhombi
Circles
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Using a min-max strategy for area of a convex set containing a segment, a triangle and a rectangle to show a lower bound of 0.232239 for a convex cover
A possible solution has the shape of a rhombus with vertex angles of 60 and 120 degrees (π/3 and 2π/3 radians) and with a long diagonal of unit length
The problem asks for the region of smallest area that can accommodate every plane curve of length 1
The problem asks for the region of highest area that can accommodate every plane curve of length 1
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