# Mystic Rose

## Problem

Mystic Rose printable worksheet

Printable resources including circle templates

**A Mystic Rose is a beautiful image created by joining together points that are equally spaced around a circle.**

Move the sliders below to see how a Mystic Rose can be constructed. You can change the number of points around the circle.**Can you describe how to construct a Mystic Rose?**

Alison and Charlie have been working out how many lines are needed to draw a 10 pointed Mystic Rose.

Alison worked out $9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45$.

Charlie worked out $\frac{10 \times 9}{2} = 45$**Can you explain how each method relates to the construction of a 10 pointed Mystic Rose?**

How would Alison work out the number of lines needed for other Mystic Roses?

How would Charlie work them out?

Whose method do you prefer?**How many lines are needed for a 100 pointed Mystic Rose?**

Could there be a Mystic Rose with exactly 4851 lines?

Or 6214 lines?

Or 3655 lines?

Or 7626 lines?

Or 8656 lines?

How did you decide?

**Final Challenge**

In a chess tournament every contestant is supposed to play exactly one game against every other contestant.

However, contestant A withdrew from the tournament after playing only ten games, and contestant B withdrew after just one game.

A total of 55 games were played.**Did A and B play each other?**

*You may wish to try the problems* Picturing Triangle Numbers *and* Handshakes*. Can you
see why we chose to publish these three problems together?**You may also be interested in reading the article* Clever Carl*, the story of a young mathematician who came up with an efficient method for adding lots of consecutive numbers.*

## Getting Started

Now look at the completed mystic rose - can you describe how many lines are drawn from each point?

How do these relate to "Alison's method" and "Charlie's method"?

## Student Solutions

Several students from Orchard Junior School noticed how the mystic rose was being constructed:

I found out that the first point connects to all of the points, then the rest of the points went down 1 until there is none to connect.

On a 6 point mystic rose, you would need to add 5, 4, 3, 2 and 1 to make how many lines there are.

6+5+4+3+2+1 for a 7 point one, and so on. All the answers are triangular numbers.

I discovered that the first set of lines that you do will have 1 less than the amount of points in the mystic rose and then you just keep on taking away one.

I worked out that every number of lines that come up at the same time decreases by one each time. e.g 7+6+5+4+3+2+1

Thomas from Wilson's School wrote:

Both Alison and Charlie are right with their calculations.

Alison would work out other mystic roses by starting with (n-1) lines and then decrease by 1 each time and adding them up. Charlie would use the formula. They should always get the same result.

The advantage for Charlie is the formula is a lot quicker to do but Alison's method is more obvious.

Lucie from Munich came to the same conclusion.

James from Wilson's School agreed that both Alison and Charlie were right:

Alison's theory relates to the diagram because each point has to join every other point once, so if we consider a 10 pointed mystic rose, the first point connects to all the other 9 points, the second joins up with 8 points (because it's all ready joined with the first), etc, etc.

Charlie's method relates to the diagram in the way that there are 10 points around the circle, and each one must be connected with a line to the other 9. But he must not count every line twice, so after multiplying 10 by 9, Charlie divides by 2.

Alison would work them out by taking the number of points on the circle, then adding all the consecutive numbers before that number (not including the number of points on the circle). Charlie would work them out by, every time, multipling the number of points on a circle with one less than its number, then dividing the outcome by 2. They will always get the same result.

A student from Mearns Castle explained this well:

Allison and Charlie are both correct (both equal 45).

It seems that Alison saw that n-1 lines were drawn from point 1, n-2 lines were drawn from point 2 and so on, which in this case is 9+8+7+6+5+4+3+2+1.

It seems Charlie looked at the completed mystic rose, and saw that there were n-1 (9) lines drawn from each of the n (10) points, so multiplying them together would give the right answer. However, this includes each line twice, thus you must divide by 2.

1+2+3+4+5...+(n-1) = n(n-1)/2 so they will both be right for any case.

Jonathan from Wilson's School agreed:

Alison and Charlie are both right, and their methods relate to the diagram:

Alison is counting the lines from 9 lines at the start to 8 then 7 and so on, whereas Charlie has devised a method which counts the points and the amount of "first lines", times them and then divides by two.

For a mystic rose of 25 points Alison would go 24+23+22+21+20+19+18...until she got to 1. Whereas Charlie would do 25x24 divided by 2, so they will always get the same result, but Charlie's method is quicker.

For a 100 pointed mystic rose you need 4950 lines as 4950 = 100x99 divided by 2.

The advantage of alternative methods is that you can compare them to see if you always get the same result and if that answer is right.

Craig wrote:

Alison and Charlie have two working methods which will correctly show the number of lines for any completed mystic rose.

Charlie's has an actual mathematic formula which allows faster calculation of the answer, and is what is used by mathematicians. As the number of points increases, Alison would take longer to work out the number of lines, as the number of calculations needed will increase. Charlie would simply be working with two larger numbers.

Janahan, also from Wilson's School, connected the number of lines to triangle numbers:

The way in which you know how many lines you need for every mystic rose depends on how many points it has.

If it has 5 points then you find the 4th triangular number.

If the mystic rose has 12 points then you find the 11th triangular number.

Example: 5 pointed mystic rose: 4+3+2+1=10 and 10 is the 4th triangular number.

Esther from St Bede's interchurch school in Cambridge used these results for the final part of the problem:

Mrs Dillon's class from Ashville College in Harrogate agreed:

Similar solutions with similar good explanations were sent by Matthew and Amar from Wilson's School and Adam from Bradon Forest School. Well done to you all.

Jack from Athol Road Primary School in Springvale South, Victoria, Australia sent us the solution to the Final Challenge about a Chess tournament. Here is his solution:

## Teachers' Resources

### Why do this problem?

This problem offers students an opportunity to relate numerical ideas to spatial representation, and vice versa.

### Possible approach

*These printable resources may be useful: Mystic Rose (worksheet),*

Circle templates.

Circle templates.

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 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.

### 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?