Subject: Paulos on problems
Date: Mon, 02 Dec 2002 16:35:53 -0500
From: Scott Williams <firstname.lastname@example.org>
Here's an article you might find interesting.
Unsolved Math Mysteries
Math Problems from the Clay Classics
to Chomping Cookies
By John Allen Paulos
Special to ABCNEWS.com
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, Poincare 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
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
To see that the number is at most 6 imagine that you're one of 6 attendees
at a conference. Since you either know or don't know each of the other 5,
there are at least 3 of them that you know or at least 3 of them that you
don't know. Suppose that you know 3 of them (the argument is similar if there
are at least 3 you don't know) and consider what relationships hold among
your 3 acquaintances.
If any 2 of them know each other, then these 2 and you constitute a group
of 3 mutual acquaintances. On the other hand, if none of your 3 acquaintances
know each other, they constitute a group of 3 mutual strangers. Thus 6 attendees
are sufficient, and it can be shown that 5 are not. (Can you find an example
showing that 5 attendees are not enough?)
Now for harder cases. It's tricky, but it can be shown that a group must
contain 18 attendees in order for there always to be at least 4 people in
it who are mutually acquainted or at least 4 who are mutual strangers. It
is still not known, however, how many people a group must contain in order
for there always to be at least 5 people in it who are mutally acquainted
or at least 5 who are mutual strangers. (The answer has been shown to lie
between 42 and 55.) Despite supercomputers, almost nothing is known about
this phenomenon for numbers larger than 5.
How to Avoid the Poisoned Cookie?
The third unsolved problem involves a game called Chomp, invented by mathematician
David Gale in 1974. Chomp has very simple rules, but no one has yet found
a winning strategy for it that always works. The game requires you to consider
a rectangular array of cookies (in rows and columns) in which the cookie
in the lower left corner is poisoned. Players A and B must take turns picking
any cookie they wish and then eating it as well as any cookies in the region
to the right of it and above it. The player who is forced to eat the poisoned
Illustration of the starting position and possible first bites for A and
OOOOOOOOOO OOOO OO
OOOOOOOOOO OOOO OO
OOOOOOOOOO OOOO OOOO
OOOOOOOOOO OOOOOOOOOO OOOOOOOOOO
POOOOOOOOO POOOOOOOOO POOOOOOOOO
It's easy to prove that there exists a winning strategy and that, if he knows
it, the first player A can be guaranteed a win. Except in
special cases, however, what that strategy is remains unknown.
One such special case occurs when the cookies are in a square array. Here
A can choose the cookie diagonally up from the corner poisoned cookie and
eat it and all cookies to the right and above it. This leaves an L-shaped
region with an equal number of cookies in each of its two arms.
A's square strategy:
Then whatever number of cookies B chooses to eat from one arm, A chooses
to eat the same number from the other arm. This continues until B is forced
to eat the poisoned cookie. Easy in this case, but a general winning strategy
that will always work with, for example, 17 rows and 31 columns is not known.
Unfortunately, my $1 million offer for a solution to any of these problems
isn't payable until the second Tuesday of the week it's published.