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## A Relook at Unsolved Problems...20+ Years Later

I enjoy finding old books or magazines that contain predictions or prophecies, and then checking on what happened. That is, was the author a seer (i.e. correct) or a sham (i.e. incorrect)?

The same can be done in mathematics, but in a different way. Consider Victor Klee and Stan Wagon's text Old and New Unsolved Problems in Plane Geometry and Number Theory, published in 1991. More than 20 years have passed...How many of these problems have been solved?

The nice thing is that many of the posed problems are not only explainable to secondary students, but also understandable. For example, consider this geometric poser: Does every simple closed curve in the plane contain all four vertices of some square?

An example of when it does occur: Do you think this problem has been solved? Draw some closed curves and investigate it yourself...trying to draw one that does not "fit" a square...now, be careful, as you might get caught up in the problem!

Or, consider three more of the other "understandable" problems:

• Does there exist a polygon that only tiles the plane aperiodically? (Note: A pair of the Penrose tiles do...but is there a single one...)
• Does there exist a box with the lengths of sides, face-diagonals, and main diagonals all integers?
• The 3n+1 problem: Is every positive integer eventually taken to 1 under iteration of the function f(n) = n/2 (if n is even) and f(n) = 3n+1 (if n is odd)?
Now. admittedly, some of the posed problems have received a lot of press in the last 20 years. An example is Fermat's Last Theorem and its solution by Wiles.

But, as a fun exercise, take each problem in the book, and use the internet to try to discover if it has been solved. For example, relative to the closed curve/square problem, consider Mark Nielsen's website (a University of Idaho mathematician). He includes a lot of known and still unknowns relative to the problem.

If a particular problem has not been solved, try to determine what progress has been made on it? And, can you locate some new problems that have been recently posed that should be in an updated version of this book?