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# Mystic Rose

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### Just Rolling Round

### Coke Machine

### Just Opposite

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 14 to 16

Challenge Level

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.*

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?