This is the summary of a presentation given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

We examine a connected series of four problems in elementary number theory that are ideal for discovery learning at several levels. Each problem generates interesting questions and conjectures, and their surprising connections add interest and allow each to shed light on the others. That some questions are open adds challenge.

1. Carousel Numbers
   The number N = 142857 has a neat property: when you multiply it by 2 through 6 you get the same digits in the same order but cycled around. 5N = 714285 for example. The next such "carousel number" we know is 0588235294117647. Multiply it by 2 to see why we need the leading 0. We know how to generate lots more, but no one knows if there are an infinite number of them.

2. Periods of Decimals
   That decimals for rational numbers are periodic is well known. It is surprising that we know little more than Gauss did about what determines these periods. The decimal for 1/7 is 0.142857142857... and has period 6. That 1/7 has maximal period and 1/11 = 0.090909... doesn't hints at the connection with carousel numbers.

3. One Numbers
   They are 11, 111, 1111, etc. Their prime factorizations have nice patterns. They illuminate problem 2.

4. The Group of Units mod n
   We use the power of group theory to get more answers and prove some of our conjectures.

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Dr. Gary Klatt
Dept of Mathematics and Computer Science
Univ of Wisconsin - Whitewater
Whitewater, WI 53190
email: klattg@uwwvax.uww.edu