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?

Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

The Perforated Cube

Stage: 4 Challenge Level:

Edison from Shatin School included some edited versions of the diagram given in the hints to support his argument:

The most is 41 blocks, as is the picture in hints. Every block you try and add will change of of the faces. So the maximum is,

Then you can take away blocks, checking each face projection so its unchanged.
On the far E, you can take away 4 on the top prong, 4 on the bottom prong, and the 1 back block on the middle prong. The middle of the S cannot be removed as it is needed for the S face. The on the close E you can take 4 from the middle prong, and then the back block on the top and bottom prong.
So we have removed $15$ blocks, and you cannot remove any more. So the minimum total is $41-15=26$

Well done Edison, can anyone think of any other interesting projections to aim for?