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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.

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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.

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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.


Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In all three cases find the number of cycles that are repeated to give the complete path and deduce the total turn for each cycle.

Use Logo to give the instructions for drawing the pattern. Design some patterns of your own.

You do not need to have met the Logo programming language previously in order to do this question. You can download the free MSWlogo program from

If you have not used Logo before then, from the first part of this simple introduction to Logo for beginners, you can learn enough in five minutes to use Logo to experiment in drawing the spirolaterals.