Wrapping Puzzles
by Serhiy Grabarchuk -- serhiy.g (at) gmail.com

I made a list of different wrapping puzzles. There are many possible variations, but I've chosen just those about simplest regular shapes and solids. For all of them there is a task to find the smallest paper polygon/shape to fully wrap a given flat (and supposed to be very thin) regular polygon/shape or regular polyhedron/solid. All wrapped things have unit sizes. An overlap/overhang of paper is allowed, but no cut is permitted.
Tasks 1 through 24 below refer to the respective diagrams 1-24. In these diagrams wrapped shapes/solids are shown in green. The meaning of letters in the diagrams is the following: a - side/edge of a wrapped shape/solid; d - diameter of a wrapped circle; A - side of a wrapping polygonal shape; D - diameter of a wrapping circle.

Wrapping of 2-D shapes
1. Wrap an equilateral triangle in an equilateral triangle. 2. Wrap an equilateral triangle in a square. 3. Wrap an equilateral triangle in a regular hexagon. 4. Wrap an equilateral triangle in a circle.	Best Known Answers
5. Wrap a square in an equilateral triangle. 6. Wrap a square in a square. 7. Wrap a square in a regular hexagon. 8. Wrap a square in a circle.	Best Known Answers
9. Wrap a regular hexagon in an equilateral triangle. 10. Wrap a regular hexagon in a square. 11. Wrap a regular hexagon in a regular hexagon. 12. Wrap a regular hexagon in a circle.	Best Known Answers
13. Wrap a circle in an equilateral triangle. 14. Wrap a circle in a square. [4] 15. Wrap a circle in a regular hexagon. 16. Wrap a circle in a circle.	Best Known Answers
Wrapping of 3-D shapes
17. Wrap a regular tetrahedron in an equilateral triangle. 18. Wrap a regular tetrahedron in a square. 19. Wrap a regular tetrahedron in a regular hexagon. 20. Wrap a regular tetrahedron in a circle.	Best Known Answers
21. Wrap a cube in an equilateral triangle. 22. Wrap a cube in a square. [1, 2, 3, and 4] 23. Wrap a cube in a regular hexagon. 24. Wrap a cube in a circle.	Best Known Answers

Part of the above tasks make a nice and fun challenges, sometimes very tricky. Some puzzles have trivial solutions or look such. The others pose rather complex optimization problems; for example, 2-D challenges with wrapping a square in a regular hexagon, and wrapping a regular hexagon in a square, or 3-D challenges of wrapping a regular tetrahedron in a square, and wrapping a cube in an equilateral triangle (or in a regular hexagon).

Some wrapping puzzles either were discussed in general, or even solved already; a few of them (with their results) were described so far [1, 2, 3, and 4]. Still I'm not aware even about any results for many of these puzzles. In all such cases presented solutions are mine. In some of these cases there are several possible wrapping patterns which, however, keep the resulting ratio for sizes of wrapping-wrapped shapes(solids) intact. Who can come up with better results?

An additional remark. For all 3-D wrapping patterns it's possible to use them to form respective 3-D solids using respectively creased paper shapes just like in Origami; see, for instance, two versions of an Origami Cube described in [3]. Same is true and for some of the above 2-D wrapping patterns. In 2-D cases you're able to form flat, package-like Origamic shapes with one plain color at both of their sides, assuming that wrapping paper is colored on one of its side and white on the other. Note, that if you use the above patterns in the Origamic mode, then shapes/solids formed of any given paper shape will have maximal possible sizes.

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Some sources on the matter.

1. Martin Gardner, Mathematical Magic Show, Alfred A. Knopf, New York, 1977, Chapter 5 - The Papered Cube puzzle.

2. Wrap a Unit Cube - proposed by Derek Bosch; originally posed (with a tiny difference) by Martin Gardner in [1].
http://www.mathpuzzle.com/wrap.htm

3. Origami Cube - a webpage by Jurgen Koller.
http://www.mathematische-basteleien.de/oricube.htm

4. Wrapping Polyhedra - by Erik Demaine.
http://theory.lcs.mit.edu/~edemaine/wrapping/


May 28, 2004