### Why do this
problem?

This problem gives scope for investigation, spotting patterns,
working systematically to cover all cases and making and proving
conjectures. It provides an example of the mathematics of

dynamical systems. This is
an important subject in higher mathematics and, in this problem,
learners can work with whole numbers in a simple discrete system to
discover for themselves the important concepts of cycles and fixed
points.

### Possible approach

Ensure that the learners understand how the mapping works then
suggest that they choose their own starting numbers and work out
their own sequences individually, making notes of anything
interesting that they observe. They might need to spend time
developing a sensible recording system to prevent confusion with
the numbers at each step. After about 10 minutes ask the learners
to work in pairs and explain to each other what they have
discovered. Then later have a class discusion to compare findings
from the whole class.

### Key questions

What happens to the sequences?

Will they go on for ever? Why?

What patterns do you notice? Can you explain them?

Do sequences have the same behaviour for ALL 2 digit starting
numbers? Why?

### Possible extension

Investigate the problem for sequences starting with negative
numbers or 3-digit numbers or bigger numbers. In what circumstances
might fixed points arise? Can students invent similar systems for
themselves?

### Possible support

Suggest that students start off with the concrete cases $a=b$ for 2
and 3. Then ask what they expect to happen for 88 and 99. Then try
it out. Were they correct?

Happy Numbers is a similar problem which lends itself to
investigation

using spreadsheets.
For a full discussion of some simple discrete dynamical systems
see:

Whole Number Dynamics I
Whole Number Dynamics II
Whole Number Dynamics III
Whole Number Dynamics IV
Whole Number Dynamics V.