Difference Dynamics
Good work on this investigation came in
from Patrick of Woodbridge School; Fred of St
Barnabas; Will, Todd, Dan, Alfie and Chrissie of Colyton
Grammar School; Niharika of Leicester High for Girls and
Damini Grover of NewStead Wood School For Girls.
Nobody actually proved that the process will always end by
repeating an earlier pattern, or in other words that it is
impossible to get an infinite sequence of triples with no
repetitions.You might like to try to prove this, it's not
difficult, and then please submit your proof. The results already
submitted are given below. There are other discoveries yet to
be made and results to be proved in this investigation. How about
the simple case of seqences starting with 2 numbers? Let us
know what you find out.
This process is an example of a Dynamical System. It is a
particularly simple example as only whole numbers are involved but
it exhibits typical patterns for the iteration converging to a
fixed point or a repeating cycle. The study of Dynamical Systems is
an important branch of mathematics.
All observed that the process for triples seems to stabilise at
$\{x, 0, x\}$ so that when one zero occurs the iteration gives a
cycle of three triples over and over again indefinitely, that
is
$$\{x, 0, x\};\{ x, x, 0\} ; \{0, x, x\}.$$
Will, Todd, Dan, Alfie and Chrissie called this 3-cycle an end triangle, a good name
for it. They observed that the number $x$ which occurs in
this 3-cycle depends on the highest common factor of the three
numbers at the start and if all the numbers in the original
triangle are the same the end triangle will be all zeros. Niharika
noted this and gave examples that starting with $\{42, 38, 8\}$
gives $x=2$, starting with $\{17, 28,41\}$ gives $x=1$ and
starting with $\{15, 10, 5\}$ gives $x=5$. Damini explored the
sequences arising according to whether the numbers at the start
were even or odd.
If the iteration gives $\{0, 0, 0\}$ then this is a single fixed point for
the iteration so we see two possible results for the iteration of
sequences of 3 numbers: (1) a 3-cycle and (2) a single fixed
point.
Niharika noted that her feeling is that if there are 4 numbers in
the original shape the numbers will always be all zeros at the
end, in other words all these iterations seem to end in the single
fixed point $\{0, 0, 0, 0\}$. She remarked that with 5 numbers
the patterns seemed to be similar to the triples and in this
case the iteration ends in 5-cycles but she could not find a
general rule for iterating sequences of 5 numbers.
Patrick noticed that this system seems to be somewhat similar to a
method for finding the GCD of two numbers, the Euclidean algorithm
and he wrote: "With this in mind, along with the fact that the GCD
is equivalent to using modular arithmetic with the larger number
mod the smaller number, I discovered that, starting with $\{a, b,
c\}$, the system seems to follow a set pattern: the output
pattern stabilises to ($f(a)$ mod $ (b-c)) + 1$ for some function
$f(a)$ which I couldn't determine immediately, but which always
seems to output a divisor of $ (b-c)$. After some more
experimentation, I found that $\{a,b,c\}$ stabilisation is closely
linked to whether $ (b-c)$ is coprime to $ (a-c)$. I derived this
within Mathematica; I attach the
notebook as a PDF for anyone who has access to Mathematica, but
I realise this isn't everyone. "