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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

10 icni Acdm Revie MATHEMATICAL PROBLEMS IN YOUR LIFE 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 "dried up" 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 "solve" 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 104 Wisconsin Academy Review

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