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 Problem set 1, problem 2 manylayers version.
 What is the minimum number of creases needed to be removed to make a crease pattern flatfoldable?

L02

 Pseudopolynomial upper/lower bounds for strip method of folding anything.
 Characterize possible seam placements.

C03

 Characterize singlevertex flatfoldable 3D crease patterns.

L04

 Optimal wrapping of other shapes by a square.
 Optimal wrapping of a cube with an x × y rectangle of paper.
 Do there exist other optimal wrappings of a cube by a square?
 Lower bounds for checkerboard folding.

C04

 Optimal 2×2 checkerboard folding.

C06

 For sufficiently small, rigid motion, is local foldability enough?
 Computational complexity of determining rigid foldability of crease patterns.
 Can a paper shopping bag be unfolded from the flat state by adding extra creases?

C07

 Universal folding of polyhedra other than boxes (e.g., polyoctahedra).
 Is there a simpler proof of flatfoldability NPhardness?
 3×n map folding. [Hard]

L08

 Prove a lower bound on number of creases in foldandcut related to local feature size.
 Higher dimensional foldandcut.
 Instantaneous flattening of polyhedral complexes.
 Connected configuration space of polyhedral piece of paper?
 Prove conjectures about linear and circular corridor density.

C08

 Foldandcut with arcs of constant curvature.
 Can we continuously flatten nonconvex polyhedra?
 Prove conjectures about linear and circular corridor density.

L09

 Do triangulated creases for hypars exist for all numbers of pleats and angles?
 Do circular pleats exist? [Hard]
 What is the maximum volume whose surface is a folding of a teabag.

C09

 What creases work for regular kgon pleats?
 Tight bounds for 1D pleat folding (allowing unfolding).
 Find an explicit example of a 1D M/V pattern which requires Ω(n/lg n) folds.
 Computational complexity of finding the shortest fold sequence to produce a given 1D M/V pattern (allowing unfolding).

L10

 Characterize when there are folding motions for paper with holes.
 Does adding a finite number of creases suffice to allow a folding motion between two folded states if the target folded state does not touch itself?

L11

 Develop a faster 2D rigidity testing algorithm, or prove a lower bound. [Hard]
 Characterize generic 3D rigidity. [Hard]

L13

 Prove lower bound relating to feature size on number of steps to unfold polygon.
 Improve step bound for energy method to unfold polygon.
 Is there a unique minimumenergy configuration of a polygon?
 Are there nonlinear locked trees of less than 8 bars?
 Characterize locked linear trees.
 Is there a locked equilateral anything in 3D?

L14

 Are there nonslender adornments that never lock?

C14

 5D and higher dissections.
 Efficient algorithm to check for matching Dehn invariants.
 Any algorithm to find a dissection when one exists.

L15

 Edge unfolding convex prismatoids.
 General unfolding polyhedra. [Hard]
 Can the star unfolding (or other edge/general unfoldings) be continuously bloomed?
 Edge unfolding a convex polyhedron into o(F) parts?

C15

 Does inverted sun unfolding (source/star) avoid overlap?
 Does every Johnson solid have an edge zipper unfolding?
 Does every convex polyhedron have a general zipper unfolding?
 Which triangulated polyhedra are ununfoldable after attaching a witch’s hat to each face?
 Are 12face polyhedra unununfoldable?
 Can prismatoids or even prismoids be fully band unfolded?
 Continuous blooming of star unfolding, sun unfolding, all edge unfoldings, all unfoldings, or orthogonal polyhedra.

L16

 Vertex unfolding convex polyhedra. [Hard]
 Grid unfolding orthogonal polyhedra. [Hard]

C16

 Convexfaced vertexununfoldable polyhedron.
 Unfolding hexagonal polyhedra.

L17

 Prove dependence of algorithms for Alexandrov’s Theorem on feature size.

C17

 Algorithm for BuragoZalgaller Theorem guaranteeing nonconvex polyhedron for any gluing.

L18

 Complexity of whether a polygon of paper can be glued into a convex polyhedron.

L19

 Which polyhedra have common unfoldings?
 Are there two polycubes with no common grid unfolding?
 Close the genus gap for nonorthogonal polyhedra with orthogonal faces.
 Minimum perimeter (and area) folding of a sphere.

L20

 Complexity of 3D min/max span.
 Flatstate connectivity of open chain, orthogonal tree, etc.
 Locked equilateral equiangular fixedangle chain?

L21

 PTAS or APXhardness for optimal folding in HP model?
 Unique foldings in nonsquare HP model?
 Minimum number of cuts to unlock an nbar open chain?
 Smallest kchain that interlocks with a 2chain?

O21

 Complexity of shortest flip sequence.
 Maximum number of flipturns.
 Characterize infinitely deflatable polygons.
