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## 'Stars' printed from http://nrich.maths.org/

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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 hit all
points.

How else can you hit all points?

When a circle has $9$ dots you can hit all points in $6$ different
ways.

What step sizes allow you to do this?

Now consider $10$ points. Can you find the $4$ different ways in
which we can hit them all?

Explore what happens with different numbers of points and different
step sizes and comment on your findings. How can you work out what
step sizes will hit all the points for any given number of
points?

Now consider $5$ points. You can hit all points irrespective of the
step size. What 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.