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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the cogwheel A as the wheels rotate.


Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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.