# Math 611: EXPERIMENTAL MATHEMATICS Fall 2004 (Rutgers University) Webpage

http://www.math.rutgers.edu/~zeilberg/math611.html

Official name: `642:611 Topics in Applied Math (Experimental Math)'.

## GREAT NEWS RELEASE, Nov. 19, 2004

This course is now part of the Canon! No longer an ad-hoc "topics" course. Thanks to our visionary grad-chair, Professor Chuck Weibel, [read his message], it will, for ever-after, be a named course called "Experimental Mathematics" and even has an eponymous number 640:640 (creatively named by Prof. Weibel), so it could be nick-named "640 squared". Notice the significance of the 640 prefix! It is a pure, rather than an applied, math, course. Indeed it is not mathematics for experimenters but rather experimentation for mathematicians.
Last Update: Dec. 18, 2004.

TEXT: The Maple Book by Frank Garvan (Chapman and Hall) and handouts.

• Teacher: Dr. Doron ZEILBERGER ("Dr. Z")
• Classroom: Allison Road Classroom Building [Busch Campus], Room 116 [Inside computer lab].
• Time: Mondays and Thursdays , period 3 (11:30am-12:50pm)
• Dr. Zeilberger's Office: Hill Center 704 ( Phone: (732) 445-1326)
• Dr. Zeilberger's E-mail: zeilberg at math dot rutgers dot edu
• Dr. Zeilberger's Office Hours: MTh 10:30-11:00.

## Outline

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in this direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they'll decide to do research in.

We will first learn Maple, and how to program in it. Then we will learn how to design and conduct mathematical experiments, that often lead to completely rigorous proofs.

Problem Set 1, due Nov. 29, 2004.

## Maple Programs Done in Class

GOLAY (bug fixed!)

BF (Oct. 18, 2004, the Knock 'm Down game studied by Arthur Benjamin and Matthew Fluet. )

AL (Oct. 21, 2004, the Amitsur Levitski Theorem)

POLF (Oct. 25, 2004, Polynomial fitting a list)

RUIN (Oct. 25, 2004, Gambler's ruin)

GaussQ (Oct. 25, 2004, Gaussian Quadrature).

LEGENDRE (Oct. 28, 2004, Legendre polynomials from scratch).

LARA (Nov. 1, 2004, guesses rational functions, courtesy of Lara Pudwell)

ORTHO (Nov. 1, 2004, gueses orhthogonal polynomials)

VATTER (Nov. 4, 2004, Restricted permutations)

PNP (Nov. 8, 2004, Straight Line Programs in order to prove that P IS NOT NP).

FLT (Nov. 11, 2004, Fermat's Last Theorem).

Eric Rowanld found a polynomial family of solutions of a^3+b^3-c^3=1! (see his message). Unfortunately, he has been scooped by the famous number theorist Kurt Mahler, who found it in 1936 (see Davnetport's book "The Higher Arithmetic" (Dover, 1983 [originally published by Harper&Brothers, 1960] p. 164).

GuessM (Nov. 15, 2004, GuessM).

Gchar (Nov. 15, 2004, Gchar).

POLFnew (Nov. 18, 2004, new version of POLF).

GcharNew (Nov. 22, 2004, Expanded version of Gchar).

TomJohnson (Nov. 29, 2004, Tom Johnson's Perfect Rythmic Tilings) .

Added Nov. 30, 2004: I learned about composer Tom Johnson's Perfect Rythmic Tilings from Jean-Paul Delahaye's fascinating article in Pour La Science, Novembre 2004. It is described in Tom Johnson's lecture.

Lara Pudwell, ran Tom(n,4), for n=1,2..., and found a new TomJohnson configuration (with n=15), the first ever with k=4. Stand by for Tom Johnson's musical composition based on Lara's discovery. See Lara's message.

Added Nov. 30, 2004: Sujith Vijay has interesting ideas of how to prove that TomJohnson configurations exist for all k and sufficiently large n. Read Sujith's message.

RandWalk (Nov. 29, 2004, Random walks) .

CRAMER (Dec. 2, 2004, Cramers's rule from scratch)

HILL (Dec. 2, 2004, Hill diagrams, a.k.a. trees).

DTrees (Dec. 5, 2004, Tin Bian's program for computing the number of Rooted (directed) labelled trees).

JacobianConjecture (Dec. 9, 2004, playing with the still open Jacobian Conjecture and the recently closed Tame Generator Conjecture).

ASKEY (Dec. 13 2004, positivity of Taylor coeffs. of rational functions in many variables).

## Comitted Projects

• Tian Bin: Find a proof of FLT (at least for n=3) using the approach outlined in my paper. Real Analysis is Degenerate... .
• Sam Coskey and Lara Pudwell: Write a C version for the Tom Johnson Configurations .
• Yi Jin: Discover empirically, and possibly prove, analogs of the Amitsur-Levitski Theorem for symmetric, anti-symmetric, and even quantum matrices.
• Chris Mesterharm, Random Walks.
• Mohamud Mohammed: q-Zeilberger
• Chris Ross: End Game of Backgammon. Write a program that inputs a Backgammon position (with the pieces no longer able to capture), and outputs the expected number of moves to the end, and the probability of each of the players to win, using the greedy strategy. Try to find closed-form formulas for general backgammon (with arbitrary number of pieces), and general die (possibly loaded).
• Eric Rowalnd: Do experiments with variation on FLT, n=3.
Posted March 1, 2005. Here are Eric Rowland's FLT project and accompanying tables, compiled by Koyama, and corrected by Eric Rowland .
• Sujith Vijay: Study the Take 'Em Game of Arthur Benjamn and Matthew Fluet. Download Sujith Vijay's Message and Maple Code

## Untaken Projects

• Write a program that plays, and studies, Renju (generalized TicTacToe) with a k by k board, and r-in-a-row.
• Write a Maple program that automatically generates a Calculus I exam with `nice numbers' for each and every kind of problem that showed up in the last five years. It should also automatically generate the answers.
• Write a Maple program that automatically generates a Calculus II exam with `nice numbers' for each and every kind of problem that showed up in the last five years. It should also automatically generate the answers.
• Write a package for Coding Theory.
• Write a package for Graph Theory, verifying empirically some famous theorems.
• Find out how you and your computer could have discovered Kathy O'hara's beautiful (KOH), from scratch, at least for small values of k.