Iteration #1.
An infinite sequence of integers A, B, C,
D, ... is generated as follows:
Given positive or negative integers A and B,
if A + B is even, then C is
the average of A
and B. If A + B is odd, then C
is the difference B - A. Now the same
process is applied to B and C to obtain D, and so on.
What is the terminal or steady-state behavior of this
sequence as a precise formula involving only A
and B? The first 207 terms are displayed
from left-to-right starting at the top, going down the
rows in the table, below. For
example, with the choice of A and B,
below, the terminal behavior of the sequence is an
unending cycle. This precise cycle could have been
predicted just by knowing A and B.
Experiment by highlighting the right-hand sides and
typing other positive or negative
integers for A and B.
Iteration #2. An
infinite sequence of positive integers A, B,
C, D, ... is generated as
follows: Given positive integers A and B,
if A + B is even, then C is
the average of A and B. If A + B is odd, then C
is the average of the absolute difference |A - B|
and 1, in other
words C equals (|A - B| + 1)/2.
Now the same process is applied to B and C
to obtain D,
and so on. What is the terminal or steady-state behavior
of this sequence as a precise
formula involving only A and B?
The first 153 terms are displayed from left-to-right
starting at the top, going down the rows in the table,
below. For example, with the choice
of A and B, below, the limiting value
of the sequence is the number 1,052. This number
could have been predicted just by knowing A and B.
Experiment by highlighting the
right-hand sides and typing other positive integers for A
and B.
Iteration #3. An
infinite sequence of pairs of integers (A,
B), (C, D), (E, F)...
is
generated as follows: Given positive or negative integers
A and B, then C is the average
of A and B (rounded down to the nearest
integer when A + B is odd) and D
is the
difference B - A. Now the same process
is applied to the pair (C, D) to obtain
(E, F), and
so on. What is the terminal or steady-state behavior of
this sequence? It seems that the
pairs always return to the starting pair (A,
B) but there is no known proof of it! The first
207 terms are displayed from left-to-right starting at
the top, going down the rows in the
table, below. Furthermore, when you display these pairs
in a scatter plot, remarkable
symmetries appear. One of them we'll call blue
symmetry and the other yellow
symmetry (scroll down to see these). From every
pair in the sequence draw a blue
horizontal line which ends, roughly speaking, at its
reflection across the blue/green axes
through the origin. If you find another pair at
this reflection point, then the sequence has
blue symmetry. Alternatively, from every pair in the
sequence draw a yellow line
through the point (0, ½) and continue on to the same
distance as the chosen pair. If you
find another pair at this reflected point, the sequence
has yellow symmetry. Your choices
for A and B cannot be very large; when A
and B are only in the thousands, complete
orbits routinely have billions of terms!