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