Challenge Level

*Stars printable sheet
There are printable circles of dots on the NRICH Printable Resources page, under Circle templates > Without central point*

Draw some stars using the interactivity below.

Choose how many points you would like around your circle

Now choose a starting point and drag to another point. Watch as the pattern continues to travel around the circle...

When a circle has 8 dots you can move around the circle in steps of length $1$, $2$, $3$, $4$, $5$, $6$ or $7$.

If you move around the circle in steps of $2$, you miss some points

If you move around the circle in steps of $3$, you visit all the points.

How else can you visit all the points?

When a circle has $9$ dots there are $6$ different step sizes where you visit every point.

Which step sizes allow you to do this?

Now consider $10$ points. Can you find the $4$ different step sizes in which we can visit every point?

Explore what happens with different numbers of points and different step sizes.

**How can you work out what step sizes will visit all the points for any given number of points?**

Now consider $5$ points. You can visit all the points irrespective of the step size. Which other numbers have this property?

**Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?**

If you would rather work on paper, go to the Printable Resources page to open PDF files of the circles (Circle templates > Without central point)

Each PDF file contains 12 identical circles with a specific number of dots. You can select any number of dots from 3 to 24.