### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

### Rolling Around

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

# Rolling Triangle

##### Age 11 to 14Challenge Level

Brian sent us his thoughts on this problem:

What happens: It is impossible to tell from the motion of the plank that the roller is not circular. The plank moves forwards at a steady rate (assuming a steady push!) and remains at a constant height (the side length of the triangle) above the floor.

Why: Consider a starting point with C on the floor and the plank parallel to the floor and touching the roller at B (BC is vertical). Move the plank, rolling the roller, until the plank is touching the roller at A (so now AC is vertical). The roller will be rotating about C. Clearly, given that curve AB is equidistant from C, the plank will remain at a constant height above the floor. The same arguments can be used if a vertex is against the plank and the circular section is against the floor; only the frame of reference is changed and relative motion of plank and floor are the same.At the point of transition, both cases are simultaneously true.
Now consider the transition. In this example, C will be on the floor, A will be against the plank, and any further movement will result in A being effectively fixed on the plank and curve BC rolling on the floor. Note that the tangent to curve BC at C is parallel to the tangent to curve AB at A --- AC is a common radius of both curves. Hence at the transition there will be smooth transfer of movement from one mode to the other. Despite one's intuitive notions, the roller will therefore have a smooth forward motion!