### Why do this problem?

This problem builds on ideas from the problem

Vector Walk.
Students are encouraged to think geometrically about vectors in
order to deduce which vectors could generate particular sets of
points, as well as reasoning why other sets of points could not be
reached in the same way.

### Possible approach

Students could work in small groups to create some vector
walks using the vector ${\bf i} = (1,0)$ and another vector ${\bf
u}$. Encourage them to comment on the similarities
between the points reached when different vectors ${\bf u}$ are
chosen.

After students have a feel for how two vectors can be used to
reach a variety of points on the plane, ask them to sketch possible
arrangements of points around the origin forming an equilateral
triangle, and a hexagon, using points which can be reached using
the vector ${\bf i}$. Some calculation will be needed to work out
the second vector needed to reach the appropriate points.

Finally, set students the last challenge to explain why a
regular pentagon could not be created in the same way. Encourage
them to use both algebra and geometry to justify their
answer.

### Key questions

Can you draw a diagram to show an equilateral triangle or a
hexagon of points around the origin?

Can you show a way to visit all these points using just ${\bf
i}$ and one other vector?

What happens when you try to do the same thing with a
pentagon?

### Possible extension

How might the ideas in this problem relate to finding roots of
unity in the complex plane?

### Possible support

Start with the problem

Vector Walk to get
an idea of how we can find a set of points that can be visited
according to some vector rules.