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Packing problem

Packing problems are one area where mathematics meets puzzles. Many of these problems stem from real-life packing problems.

In a packing problem you are given

  • one or more (usually two-or three-dimensional) containers
  • several 'goods' some or all of which have to be packed into this container

Usually the packing requires to be without gaps and overlaps, but in some packing problems the overlapping (of goods with each other and/or with the boundary of the container) is allowed and has to be minimised. Hence we can discern several categories of packing problems:

Table of contents

Categories of packing problems

  • No gaps or overlaps allowed
  • Gaps allowed, but no overlaps. Usually the total area of gaps has to be minimised. See below for an example.
  • Gaps and overlaps allowed. Here usually the total area of overlaps has to be minimised.

Examples of 'gaps, but no overlaps' packing problems:

Example 1

This is a classical one, its outcome surprising even for many mathematicians. The problem is to fit as many circles of 1 cm diameter into a strip of 2 x n size as possible, where n = 1, 2, 3,.... Of course you can fit at least 2*n circles in there, but the surprising answer is that if n>63, then you can fit at least one more circle in than the formula 2*n suggests. Indeed, for every added length of 64, you get another additional circle in!

Example 2

How many oranges (balls) of given diameter d can you pack into a box of size a x b x c ? This is one of the hardest problems in this category.


Many puzzle books as well as mathematical journals conatins articles on packing problems.
  • Journal of Recreational Mathematics - easy read, many articles.

See also: Tetris

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