You have a rectangular solid, A x B x C in dimension, and you want to wrap it with a rectangle of wrapping paper, D x E in size.

geometry/wrapping.paper New

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Problem

You have a rectangular solid, A x B x C in dimension, and you want to
wrap it with a rectangle of wrapping paper, D x E in size.  Minimize the
area DE of the wrap.

Assumptions: The wrapping paper is not stretched, cut, or torn in any
way.

I have what I think is a pretty good answer for some classes of
rectangular solids.  My technique, however, is not a general solution,
does anyone know of one?

-- David Boll <dwboll@gr.hp.com>, November 1997

Show solution

John Rickard <jrr@stingray.atml.co.uk> wrote:

} I wrote:
} : Unfortunately, the answer is uninteresting: you can get DE as close
} : as you like to the surface area of the solid by making D very small
} : (so that the wrapping paper is a long thin strip).
}
} However, this requires folding the paper back on itself, something
} like
}
}    --------------+
}                  : \   (diagonals should be 45 degrees, of course)
}                  :  \
}    ------------------+     +
}                  :  /      | D
}                  : /       |
}    --------------+         +

No, it doesn't.  You start in one corner and *spiral* around the box
until you come to the diagonally opposite corner.  By making the paper
narrow enough you can make the 'breakage' at the very beginning and
very end as small as you want, and there will be no overlap at all over
the rest of the box.

-- Bernie Cosell <bernie@fantasyfarm.com>

Hugh ApSimon discusses a problem which is similar to this in his book
"Mathematical Byways in Ayling, Beeling and Ceiling" (Oxford University
Press, 1984).  You might want to check that to see if it is any help;
unfortunately my copy is buried in a big box with several hundred other
books, so I'm working by memory here (except for the book info - I
checked a library catalog for that).

-- Geoff Bailey <ftww@cs.su.oz.au>, November 1997