Dalmatians
Investigate the sequences obtained by starting with any positive 2 digit number $(10a+b)$ and repeatedly using the rule
$10a + b \to 10b -a$
to get the next number in the sequence.
You can take any number and write it in the form $10a+b$ , that is as a multiple of ten plus a number $b$ between 0 and 9, for example:
$$57 = 10 \times 5 + 7\quad\quad -6 = 10 \times (-1) + 4 \quad\quad 123 = 10\times 12 + 3$$
This iterative procedure is an example of a dynamical system which can be studied in more detail at university; you may read an introduction to this fascinating subject in Whole Number Dynamics 1 . Dynamical systems using decimals can have many strange and interesting properties; they form the foundation of the subject of chaos, which you can read about on the Plus website .
Thank you Alex from The Grammar School at Leeds for your solution to this problem.
All the sequences starting from 1 to 91 inclusive are 4-cycles (the first, fifth, ninth terms etc. are all equal) for example the sequence starting from 53 is: 53, 25, 48, 76, 53, 25, 48, 76, .... and so on. Alternate numbers in the sequences add up to 101 or 0.
However this is not a general rule. For example, sequences starting with numbers between 92 and 99 also go into 4-cycles but these 4-cycles start from the second term of the sequence. For example the sequence starting with 92 is: 92, 11, 9, 90, - 9, 11, 9, 90, - 9, 11, ...
The number 0 is a fixed point of this system. The problem can be generalised to apply to all integers (expressed as 10$a + b$ as above) and sequences starting from 101, 202, 303 etc. end up at 0.
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?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.