The State of Wisconsin Collection

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Scott, Walter E. (ed.) / Wisconsin Academy review
Vol. 5, No. 3 (Summer 1958)

Hammer, P. C.
Mathematical problems in your life,   pp. 104-105

Page 104

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10                           icni Acdm Revie
By P. C. Hammer, Director
UW Numerical Analysis Laboratory
Most college graduates and even most Ph.D's have the
impression that the stream of new mathematics has reduced
to a meager trickle. Now, while mathematics has &quot;dried
up&quot; to the average taste, nevertheless about 50 weighty
volumes of mathematical researches are published each year.
.Jith such a flow you might think that little will soon be
left to be done. Your courses in mathematics have prob-
ably indicated to a very limited extent the nearby and ex-
tensive boundaries of the unknown, and the kinds of prob-
lems with which mathematicians occupy themselves.
It is true that many problems depend for a reasonable
length explanation on a rather extensive technical back-
ground. On the other hand, there are mathematically formu-
latable problems which have not been solved in any realistic
sense, mathematically, but which confront you regularly.
For example, you probably have had to solve some kinds of
packing problems even if you didn't do it mathematically.
If you are able to solve the same problem mathematically,
you might be acclaimed a mathematical genius. Now it is
my purpose to illustrate a few of the classes of unsolved
mathematical problems by those which I will term the pack-
ing problem, (including the tailor's problem) and the
patching problems. They are all problems involving maxima
or minima and they have not been solved in any but a tew
special cases.
1. The packing problem. (For dishes, for apples,
for molecules.   Given one box and a set of cups and
saucers, what is the largest number of cup and saucer
pairs which can be placed in the box? This kind of prob-
lem is so difficult that it has been solved mathematically
only for very special cases such as the one in which every
object is a sphere of the same size. Yet, every person
packing objects in a limited space has to &quot;solve&quot; this
problem, and usually a much worse one, due to the variety
of cPrjects to be packed. (During July and August millions
of people will prove their fundamental mathematical ability
by packing seemingly impossible numbers of objects into
the luggage compartments of their cars). Conversely, what
is the smallest box to hold the items to be packed?
2. The patching or covering problems. These problems
may be illustrated by finding the smallest diameter of
circular patches such that, say, five of them will cover
a square area. In doing this it will probably be necessary
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