Diary for 01:090:101:21, Experimental Math, fall 2008

The tenth meeting, 11/10/2008

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.

Outline of Dr. Z.'s presentation

Dr. Z. used the word oxymoron in discussing the subject of Experimental Math. One definition of oxymoron is "A rhetorical figure in which incongruous or contradictory terms are combined ..." so 20 or 30 years ago active external (not inside the head!) experimentation in math would have been difficult or impossible. The introduction of cheap, fast, powerful computers with good software allows anyone to be an experimental mathematician.

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:

He wrote a recursive program to report if "you" can win with n pennies.

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

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.

We thanked the good doctor.

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