The PCTM Magazine

Excursions For Students

by Craig B. Merow

Is That A Problem?

Some problems, such as the Four-Color Map Problem (See this column in the PCTM Newsletter, Fall 1995.), take mathematicians over one hundred years to solve. Most problems are solved more quickly. Each year thousands of new theorems are proven, and many applied problems are solved.

During the past sixty years, Paul Erdös (pronounced AIR-dish) solved more than anyone else. With his death on September 20, 1996 an era of mathematical accomplishment ended. During his life, Erdös worked with mathematicians all over the world, often spending 16 to 20 hours a day doing mathematics. He wrote over 1500 mathematical papers, each of which contained new mathematics that he discovered.

Paul Erdös was quite a character! He had no home, no job, no bank account, no car; he didnąt even have a driver's license! He lived to do mathematics, and was content to let others take care of the details of life for him.

Paul traveled from mathematics department to mathematics department. He was known and welcomed at universities around the world. It was not unusual for him to work with mathematicians in the United States, Canada, Germany, Hungary, and Australia, all in a single month. The worldąs mathematicians were happy to take care of "Uncle Paul." He was a world treasure. He helped hundreds of young mathematics students get started in their chosen profession, and solved thousands of problems that had stumped others (Tierney 1984).

Paul was known for his ability to find short, clever solutions to very difficult problems. In 1976, while having coffee in the mathematics lounge at Texas A&M University, Paul became curious about something written on the blackboard. "What's that? Is it a problem?" Erdös wanted to know.

It was a problem, a very difficult problem in a branch of mathematics that Erdös knew little about. It had recently been solved, after much hard work, by two mathematicians. Their solution required 30 pages of calculations.

Erdös asked for some definitions of the symbols that he was not familiar with, thought for a moment, and wrote down a two-line solution (Hoffman 1987). That's brain power!

Exercises

While the following problems can be solved with elementary arithmetic, they require the same kind of insight that Paul Erdös was famous for. Give them a try; perhaps you may be the next Paul Erdös.

1. Carl Friedrich Gauss (1777-1855) is considered to be one of the greatest mathematicians of all time. When he was ten years old his school master asked the class to find the sum of the numbers from 1 to 100: 1 + 2 + 3 + ... + 100. As soon as the master had finished stating the problem young Gauss wrote the answer on his slate.

a. Gauss had immediately realized that the sum was equivalent to the product 50 x 101. Why?

b. Use the same technique to find the sum of the first 1000 counting numbers.

c. Find 2 + 4 + 6 + ... + 100.

2. Two trains are traveling toward each other on the same track. Each train is traveling at 10 mph. A bumblebee is flying back and forth between the trains as they approach each other. If the bumblebee flies at 20 mph and the trains were 1 mile apart when the bee began its shuttle, how far will the bee travel before the trains collide?

3. The circumference of the Earth is approximately 25,000 miles. Suppose a ribbon was tied around the Earth at the equator so that it fit snugly. If 20 feet of ribbon was added to the encircling band, and the new, longer ribbon was positioned so as to be equally distant from the Earth along its entire length,

a. Could you slip a piece of paper under it without touching the ribbon?

b. Could you crawl under it?

c. Could you walk under it?

4. I have two children. One of them is a boy. What is the probability that the other one is a boy?

References

Bell, E.T. 1965. The prince of mathematicians. Chap. in Men of Mathematics. New York: Simon and Schuster.

Bollobas, Bela. 1996. A life of mathematics‹Paul Erdös, 1913-1996. Focus: the newsletter of the mathematical society of America 16 (December) : 1, 4-6.

Erdös, Paul. 1985. Paul Erdös. Interview by G.L. Alexanderson (Santa Clara, December 1979). In Mathematical People. ed. Donald J. Albers and G.L. Alexanderson, 81-92. Boston: Birkhauser.

Foerster, Paul A. 1994. Algebra and trigonometry. Menlo Park: Addison-Wesley. Hoffman, Paul. 1987. The man who loves only numbers. The Atlantic Monthly, (November): 60-74.

Merow, Craig B. 1995. The four-color problem. PCTM Newsletter. 36 (Fall): 18-19. Stewart, Ian. 1977. Gauss. Scientific American. 237 (July): 122-31.
Tierney, John. 1984. Paul Erdös is in town. His brain is open. Science 84 (October):40-47.

For The Teacher

1a. Gauss found the sum, 1 + 2 + 3 + ... + 100, by adding 1 + 100, 2 + 99, 3 + 98, ... , 50 + 51, or 50 X 101. This is a special case of an arithmetic series, s(n)= n/2(t(1) + t(n)), where t(1) is the value of the first term, and t(n) the last. If no student solves this exercise, remind them of the commutative property and ask, "What is 1 + 100? 2 + 99?"

1b. 1 + 2 + 3 + ... + 1000 = 1000/2(1 + 1000)= 500,500.

1c. 2 + 4 + 6 + ... + 100 = 50/2(2 + 100)= 2550.

2. After working on exercise 1, many students will attempt to do exercise 2 with a series. This is, of course, unnecessary. The trains are approaching each other at 20 mph. Since the bee is traveling at the same speed, it will travel the same distance: 1 mile.

3. Adding 20 ft. to a circumference adds 20/pi ft. to the diameter and 10/pi ft. to the radius. Thus, most Homo sapiens could crawl, but not walk, under the ribbon.

4. 1/3

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