Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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?

Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Efficient Packing

Age 14 to 16
Challenge Level

How efficiently can you pack disks of the same size with no overlap? Imagine attempting to cover a 1m square with 10cm diameter disks with no overlap. What percentage of the area of the square can you actually cover using this obvious packing for disks?

disks image
How much more efficiently would you be able to pack 1cm diameter disks into the 1m square? Could you make an estimate for the efficiency of packing disks of diameter 1mm?

As a harder extension, make an estimate of the number of 1cm diameter balls that would be able to fit into a 1m cubed box.

Notes and background

Whilst it might seem relatively simple, the problem of 'shape packing' is often very difficult mathematically to solve with certainty for many shapes. Intuitive visualisation often works just as well as a strict mathematical analysis, and often is the only sensible possibility with packing together complicated shapes.

You might like to consider situations in which efficient shape packing is relevant in the physical world.