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Why do this problem?

This problem offers students an opportunity to relate numerical ideas to spatial representation, and vice versa.
Thinking about different ways of counting the number of lines in a mystic rose can lead to a better understanding of the general formula for triangle numbers.

Possible approach

This problem works very well in conjunction with Picturing Triangle Numbers and Handshakes. The whole class could work on all three problems together, or small groups could be allocated one of the three problems to work on, and then report back to the rest of the class.

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:
  • 4851
  • 6214
  • 3655
  • 7626
  • 8656

Key questions

What is special about the numbers of lines needed for different sizes of mystic rose?
How do the different ways of working out the number of lines relate to the construction and final image of the mystic rose?

Possible extension

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?

Possible support

Students could construct their own mystic roses using different colours for the lines from each point, to build up an understanding of their structure. Circle templates with dots evenly spaced on the circumference can be found here.