You may also like

problem icon

Whole Number Dynamics I

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

problem icon

Whole Number Dynamics II

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

problem icon

Whole Number Dynamics III

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

Difference Dynamics

Stage: 4 and 5 Challenge Level: Challenge Level:1

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. "