There are 119 colleges which play division 1A football and as of 11/9/2008 Rutgers is ranked between #64 and #70 (depending on the ranking system). There are over 4,000 degree-granting colleges in the U.S. and I suspect that most of them have math departments. For a number of years, Rutgers has been recognized (and ranked!) as having one of the top 20 (maybe even 15!) math departments in the country.

Doron Zeilberger is one of the important reasons this is so. He studies how computers and people can discover and prove mathematical statements. He is a Board of Governors Professor of Mathematics and has won a number of significant awards, including the Lester Ford Award of the Mathematical Association of American (an award for expository excellence), the Leroy Steele Award of the American Mathematical Society (an award for significant contribution to mathematical research), and the Euler Medal of the Instutite of Combinatorics, also an award for research excellence. He has worked with a continuing sequence of excellent graduate students and collaborators, and his enthusiasm and intelligence illuminate all of his mathematical activities. We are fortunate that he is the guest lecturer for today. I recommend, by the way, that you browse his web page, and especially look at his opinions, some of which are outrageous, silly, or true, and (sometimes!) all of these.

Dr. Z. used the word

**Geometry**

As an example of how accomplished "computer math" can now be,
Dr. Z. cited *Ekhad's
Geometry WebBook*, which studies and proves many of the
classical and complicated theorems of plane geometry. He specifically
mentioned Morley's
Theorem. This is a result with classical ("human") proofs that are
not too accessible, but which was proved as we watched by computer in
.45 seconds of CPU time.

**Connect 4**

We then looked at a combinatorial game. Chess is too hard, and the
standard tic-tac-toe too easy. But even Ekahd's
Connect-Four shows some interesting features. We looked at a few
of the end game challenges. Students suggested some solutions.

**Removing pennies**

Finally Dr.Z. sat at the computer and "played". He showed how the
computer could solve a simple penny removal game. Here are the rules
for the initial game:

**Starting position**a pile of consisting of a non-negative integer number of coins.**Play**Each player removes 1 or 2 coins. The player loses if there are 0 coins.

Then he varied the program, so that each player can remove 1,2,...,k pennies. We looked at that.

**Challenge**

Suppose we want to play the game where both sides can remove a number
of coins specified by a collection of integers, for example, {1,3,6}:
so either player can remove 1 or 3 or 6 coins. Write a program that
reports whether an initial pile of n coins is a win or a loss. Submit
this program to Dr.Z.
(zeilberg@math.rutgers.edu) and win $10!

**Decimal expansion**

Dr. Z. wrote a program which first found a list of decimal digits of a
number, and then the base B expansion digits of a number. He asked if
students could verify if, say, sqrt(2) had (asymptotically) the digits
of its decimal expansion distributed correctly. This is a difficult
problem, certainly worth a $1,000,000 prize!

**Some references**

The penny removal problem is a variant of Nim.

**Applause!!!**

We thanked the good doctor.

**
Maintained by
greenfie@math.rutgers.edu and last modified 11/10/2008.
**