Packing problems
Packing problems are one area where mathematics meets puzzles. Many of these problems stem from real-life packing problem.
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:
Categories of packing problems
- No gaps or overlaps allowed
- many polycube puzzles fit into this category
- many polysquare puzzles fit into this category
- many tiling puzzles fit into this category
- 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.
Example for a 'gaps, but no overlaps' packing problem:
This is a classical one, its out come 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 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!