By Thotsaporn "Aek" Thanatipanonda and Doron Zeilberger
In hindsight, we are very thankful to Jeff for his snide remarks, since this lead me to
look up his work and discover his
early paper
on Toads and Frogs, that is
admittedly a beautiful piece of work, as far as humans go. But humans can only go so far,
so our present paper is a dramatic illustration of computer-generated-mathematics
and the "ansatz ansatz".
.pdf
.ps
.tex files
[Appeared in Journal of Difference Equations and Its Applications 15 (2009), 111 - 118]
Written: Aug. 29, 2007.
This is my sweet revenge against Jeff Erickson's
snide remarks, that I have already
got mad at,
but it is better to get even. My brilliant student
Thotsaporn "Aek" Thanatipanonda,
and I taught our computer how to prove some conjectures made by that
same Erickson, way back when he was at grad school. Not only that,
it can make much more elaborate conjectures and then prove them,
all without human touch!
Important: This article is accompanied by Maple
package
ToadsAndFrogs that automatically conjectures
and then automatically proves explicit expressions for values of
certain families of Toads and Frogs game-positions.
Sample Input and Output
the
input
produces the
output .
the
input
produces the
output .
as well as the class of positions
Ta1BTa2BTa3FB
(f[2]),
as well as the class of positions
Ta1BTa2FBB (f[3]),
as well as the class of positions
Ta1FBBB (f[4]),
(what we call in the paper class A31)
the
input
produces the
output .
Added Sept. 26, 2007: I am going to talk about it at a
special session, organized by Moa Apagodu and Akalu Tefera,
on 4:30pm (local time), Jan. 8, 2008, at the AMS annual meeting
to be held in San Diego. Here is the
abstract
Doron Zeilberger's List of Papers