How to wrap packages with maximum paper efficiency #Math #ProblemSolving

Jason writes on the Almost Looks Like Work blog about the eternal problem in wrapping a package: How to do wo with the least amount of wrapping paper.

The problem:

  • The present is a cuboid of side lengths a, b, c with a > b > c
  • The wrapping paper is a rectangle of side lengths w, h
  • The wrapping process is to place the present on the centre of the wrapping paper on it’s largest face (of area ab), optionally rotate the paper by some angle \theta, then fold the paper up around the present onto the top face
  • The paper must cover all of the surface area of the present
  • The efficiency of the wrap \epsilon < 1 is defined as the ratio of the surface area of the present to the area of the paper:

\displaystyle{\epsilon = \frac{2(a\cdot b + a \cdot c + b \cdot c)}{w \cdot h}}

Our aim is to find a w, h, \theta which covers the present with maximum efficiency.

See the post for the analytical results and the conclusions.

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