Unsolvable and Unsolved Problems

Date: 02/19/98 at 18:16:57
From: D-Jitsu
Subject: 11th Grade Research Paper

Dear Dr. Math,

I need help getting started on a research paper, as well as on doing 
the project. The topic of my report is "The Impossible and Unsolved in 
Mathematics (Goldbach's Conjecture, Squaring the Circle, etc.).  I was 
just wondering if I could get some references, tips, or ideas on how 
to get started. This is for a pre-calculus class.

Date: 02/20/98 at 16:31:35
From: Doctor Martin
Subject: Re: 11th Grade Research Paper

This is a very interesting topic to explore - it would be good to 
distinguish between problems which have been proven unsolvable, and 
those unsolved. 

For instance, "squaring the circle" and the trisection of general 
angles with straightedge and compass are examples of problems which 
have been proven unsolvable.  These are problems of construction, and 
it has been proven that certain things are inconstructable. 

Goldbach's Conjecture, on the other hand, is not necessarily 
unsolvable, but just hasn't yet been solved. In other words, no one 
has proved Goldbach's Conjecture, but no one has proved that it's 
unprovable either.

I would begin with this distinction, and discuss specific mathematical 
problems, some which are unsolvable, and some unsolved.  

Unsolvable problems include the squaring of the circle, the trisection 
of general angles, solving the quintic with radicals, and the 
continuum hypothesis.  

Unsolved problems that aren't too hard to understand include 
Goldbach's conjecture, the 3n+1 conjecture, and the general Poincare 
conjecture.  

A good first place to look would be a history of mathematics book, 
such as Carl Boyer's book, _A History of Mathematics_, or 
Morris Kline's _Mathematical Thought from Ancient to Modern Times_.  
Most of the problems mentioned above are in these sources, with the 
exception of the most recent 3n+1 conjecture. There is, however, some 
material on this problem (which also goes under the name "The Collatz 
Problem") on the Web. A good starting point would be:

 http://ug.cs.dal.ca/~campbell/colllink.html   

Even though the Collatz problem is not as historically important as 
the others mentioned, it may be especially interesting for your 
discussion. First of all, it is fairly easy to understand. Second, it 
is an example of a problem that is unsolved, and there is a little 
evidence that it may be unsolvable (due to J. H. Conway).  

Even though this response is already getting somewhat long, maybe I 
should describe this problem specifically. It goes like this:

  Choose a positive integer n.  Submit it to the following process:

   (1) If n is odd, let your new number equal 3n + 1
   (2) If n is even, let your new number equal n/2
   (3) If your new number isn't equal to 1, put it back into step (1).

So for the number 7, this process yields the following sequence:
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The 3n+1 problem, or the Collatz Conjecture, states that no matter 
what number you start with, you eventually arrive at 1. Most people 
think that this is true, but it may be unprovable. This problem, I 
think, conveys the distinction between unsolved problems and 
unsolvable problems particularly well.

Good luck on your project - it's a great topic to write about!

-Doctor Martin,  The Math Forum
 http://mathforum.org/dr.math/