.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!
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 computergeneratedmathematics
and the "ansatz ansatz".
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 gamepositions.
Sample Input and Output

For the values of positions (f[1]) consisting of a_{1} Toads followed
by one Blank followed by a_{2} Toads followed by one Frog, as well
as for the values of positions (f[2]) consisting of a_{1} Toads followed
by one Frog followed by one Blank (what we call in the paper class A11),
the
input
produces the
output .

For the values of positions (f[1]) consisting of a_{1} Toads followed
by one Blank followed by a_{2} Toads followed by one Blank
followed by a_{3} Toads followed by one Frog, as well
as for the values of positions (f[2]) consisting of a_{1} Toads followed
by one Blank followed by a_{2} Toads followed by one Frog followed by one Blank
(f[2]) as well as
as for the values of positions (f[3]) consisting of a_{1} Toads followed
by one Frog followed by two Blanks,
(what we call in the paper class A21),
the
input
produces the
output .

For the class of positions of the form
T^{a1}BT^{a2}BT^{a3}B
T^{a4}F (f[1]),
as well as the class of positions
T^{a1}BT^{a2}BT^{a3}FB
(f[2]),
as well as the class of positions
T^{a1}BT^{a2}FBB (f[3]),
as well as the class of positions
T^{a1}FBBB (f[4]),
(what we call in the paper class A31)
the
input
produces the
output .

For the class A41,
the
input
produces the
output .

For the class A51,
the
input
produces the
output .

For the class A12,
the
input
produces the
output .

For the class A22,
the
input
produces the
output .

For the class A32,
the
input
produces the
output .
(This contains, amongsts many other results, the proof of
Jeff Erickson's conjecture mentioned in the paper).
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
Doron Zeilberger's Home Page