This problem offers students an opportunity to relate numerical ideas to spatial representation, and vice versa.
Start by showing the animation of the seven point mystic rose. Then reset it and ask the students to describe to their partners what they saw. Choose a different mystic rose and show the animation, pausing it as it plays. Ask the class to predict what will happen at each stage. Can they predict how many lines will be drawn in total?
Set the class the challenge of working out how many lines are needed to draw 8, 9, and 10 point mystic roses. Allow them some time to work on this. Bring the class together to discuss their answers and methods, and more importantly, how their methods relate to the construction of the mystic rose.
Set the class another challenge, this time to work out how many lines are needed to draw a much larger mystic rose (e.g. a 161 point mystic rose). Allow them some time to work on this. When they report back, discuss the need for efficient ways of working this out. Draw attention to "Charlie's method" in the problem, if no-one has suggested it, and encourage students to think about how this method relates to the image of the completed mystic rose.
For a class that has been introduced to algebra, students could express "Alison's method" and "Charlie's method" algebraically.
Finally, ask them to work out which of the following numbers of lines could be used to draw mystic roses:
Can you draw a mystic rose using 9, 19, 29, 39, ... lines? Are these impossible? How do you know?
Will there ever be a mystic rose constructed from a multiple of 1000 lines?