### Just Rolling Round

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?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Tracking Points

##### Stage: 4 Challenge Level:
Full Screen Version
This text is usually replaced by the Flash movie.

Copies of the line segment AB - shown as the yellow arrow in the animation - that joins the points (0,0,0) to (1,1,1) are linked together (end-to-end) in a chain starting at (0,0,0). What would the coordinates of the ends of the tenth segment be? You can check your answer in the hints.

If a second line segment (CD) starts at (0,0,0) and ends at (1,3,2) and copies of this are made and joined end-to-end to it, what would the coordinates of the ends of the 2nd segment be? The tenth segment? The nth segment?

What if the chain of the line segment AB described in the first paragraph started at (1,0,-1)?
Where would the ends of the nth segment be this time?
Where would the ends of the nth segment be if the first segment started at (a,b,c)?

What if copies of the second line segment, CD, started at (-3,0,2) and were placed end to end. Where would the ends of the nth segment be?
If the chain started at (a,b,c) where would the ends of the nth segment be?

If you now make lots of copies of the two line segments and decide to place them end-to-end alternately, investigate the end points of the nth segment for different starting points.