Who’s Counting?: Unsolved Math Mysteries

Dec. 1 -- Two years ago the Clay Mathematics Institute near Boston listed seven important questions in mathematics that have never been answered and offered a $1 million reward for the solution of any one of them.

Even the mere statements of these classic unsolved problems, which include the Riemann hypothesis, Poincaré conjecture, and the P-NP problem, are difficult to understand.

Happily, a just-published book, The Millennium Problems by Keith Devlin, makes them all considerably easier to grasp.

Inspired by this endeavor, I'd like here to briefly describe three unusual problems whose statements are, by contrast, quite elementary. Nevertheless, the problems have all so far resisted solution.

Why Always 4, 2, 1?

To understand the first problem, discovered by Lothar Collatz in 1937, pick any whole number you wish and subject it to the following rule. If the number is even, cut it in half, but if it is odd, multiply it by 3 and add 1. Whatever number results from this, apply the rule to it as well. If you do this over and over again, you'll notice something rather odd about the numbers that you get.

Let's check. Say you pick 13. Since 13 is odd, you multiply it by 3 and add 1 to get the number 40. Since 40 is even, you cut it in half to get 20. Applying the rule now to 20 gives you the number 10. Continuing in this way, you get 5, 16, 8, 4, 2, 1, 4, 2, 1, .... Note that if you get to 1, the 4, 2, 1 will repeat itself indefinitely.

The Collatz Conjecture is that no matter what number you choose, you'll always end up with 4, 2, 1, 4, 2, 1, ....

Sometimes it takes many applications of the rule to get there; even the small number 27, for example, takes 111 steps before it reaches 1. Every number that has been tried does eventually return to 1, but it's never been mathematically proved that every number does.

How Big a Conference?

The next type of problem may be traced to a 1928 paper by British logician Frank Ramsey and has been pursued ever since by many mathematicians. It is sometimes phrased in terms of attendees at a small conference. Let's start with a special case and ask: What is the smallest number of attendees who need to be present so that it is certain that there will be at least 3 who know each other (3 mutual acqaintances) or at least 3 who don't (3 mutual strangers)?