Moser's worm problem

The problem asks for the region of smallest area that can accommodate every plane curve of length 1.

Another possible solution has the shape of a rhombus with vertex angles of 60° and 120° and with a long diagonal of unit length.

Gerriets & Poole (1974) conjectured that a shape accommodates every unit-length curve if and only if it accommodates every unit-length polygonal chain with three segments, a more easily tested condition, but Panraksa, Wetzel & Wichiramala (2007) showed that no finite bound on the number of segments in a polychain would suffice for this test.

The problem remains open, but over a sequence of papers researchers have tightened the gap between the known lower and upper bounds.

In the 1970s, John Wetzel conjectured that a 30° circular sector of unit radius is a cover with area

1.	An upper bound is a disc of diameter equal to the length of the worm.
2.	By symmetry, half the disc is sufficient.
3.	The cover must support a width at least the worm's length divided by $π$
4.	A solution by John E. Wetzel. ^{[ 1 ]}