Difference dynamics

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

Age

14 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Choose any three numbers. The differences between your numbers give you three new numbers. Repeat this operation to give a sequence.

In this example the sequence starts: $15, 39, 8 \to 24, 31, 7 \to 7, 24, 17 \to 17, 7, 10 ...$.

What happens to this sequence. Investigate for different starting points.

What do you notice? Can you explain what happens?

How about sequences starting with four numbers, or two numbers?

See the article Difference Dynamics Discussion.

Student Solutions

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. 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. "

Teachers' Resources

Why do this problem?

This is an engaging investigation that quickly yields results and leads to conjectures and a lot of mathematical thinking and discussion. Young learners, even at the primary stage, can understand and carry out the iterative process and see the cyclical patterns that emerge. It is not difficult to make and test conjectures. Also it is easy to understand that, after the first sequence, all the sequences have only positive terms and so, by taking differences, the terms in the sequences can't increase. This means that, from any chosen starting sequence, there are a strictly limited number of possible sequences that can follow, and sooner or later a sequence must repeat itself starting a cycle. Thus it is easy to prove that all the sequences of sequences end in cycles or a sequence of zeros. Not only can very young learners create the sequences of sequences, notice the cycles and make conjectures, but also this proof is very accessible and there are more questions to explore making this a low threshold high ceiling investigation.

This is a simple example of a dynamical system and it can lead to discussion of how dynamical systems are used to model population dynamics and other natural phenomena.

Possible approach

If different groups in the class choose their own sequences from which to start, it won't be long before they notice that they are all getting the same sort of patterns. It is easier to see how the process works by starting with sequences of length three rather than length two. When everybody finds that before very long they have produced a cycle, and nobody can find a sequence that goes on indefinitely, then perhaps suggest that they try sequences of length four and see if the same thing happens. They will soon find that with length four the iteration always seems to stop with the zero sequence.

From that point the teacher can either encourage the learners to discuss why the iteration always seems to stop with a zero sequence or cycle and, depending on the class and time available, reach a well argued proof.

Alternatively the class can try starting with sequences of different lengths 2, 3, 4, 5, and 6 say, and try to discover if the lengths of the sequences determine whether the sequences go to zero or end in a cycle.

Key questions

Look at your chain of sequences, have you seen that sequence in the chain before? What will happen next?

Is it worth continuing this chain or do you already know how the chain continues?

Can you describe what is happening to your chain of sequences? (Encourage language like "it loops back on itself", don't introduce the term 'cycle' too early rather, if possible, let the term emerge in discussion).

Can the terms that occur in the sequences get bigger?

You said the terms can't get bigger so what is the biggest value any term can take in your chain? Then how many different values can the terms take? So how many different sequences is it possible to have in your chain? Can the chain go on for ever without any sequence being repeated?

Would the same thing be true for any chain?

Does the same thing happen when you start with sequences of different length?

If you get to the zero sequence what can you say about the sequence in the chain just before it?

Possible extension

Prove that when sequence $\mathbf{a}$ maps to the next sequence in the chain $\mathbf{b}$ then $\mathbf{a}$ is a constant sequence if and only if $\mathbf{b}$ is the zero sequence.

Read the article Difference Dynamics Discussion.

Possible support

Suggest the learners start with sequences of small terms which will converge very quickly.

For example: $(1, 4, 3), (3, 1, 2), (2, 1, 1,), (1,0,1), (1,1,0), (0,1,1), (1,0,1)...$

and $(1,5,3,7), (4,2,4,5), (2,2,1,1), (0,1,0,1), (1,1,1,1), (0,0,0,0)....$

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