### 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:

This problem provides a rich context for visualisation and suggests many similar lines of enquiry.

For example :
• Can all letters of the alphabet be represented using only a 5 by 5 array of cubes ?

• Is it possible to create every combination of three letters in a 'perforated cube' ?

• What difference does the orientation of the letters make ?

• How many ways are there to orientate three non-symmetric letters such as F, J and P ?

• What difference will it make to use letters that have symmetry of some kind ?

For any chosen combination of letters and orientation, explore the variation possible in successful arrangements.
• Can every solution arrangement have either more cubes added or some removed, and still be a solution arrangement ?
• Is there a relationship between the maximum and minimum number of cubes for any solution ?
Exploring the perforated cube problem, and related questions, using clip-together plastic cubes will provide invaluable 'concrete' sensory experience for students as they stretch their powers of visualisation, express the problem, test their conjectures, or represent their solutions.

The Freudenthal Institute for Science and Mathematics Education has some very useful Java interactivities.
Follow this link to access these environments : www.fi.uu.nl/rekenweb/en/

The best interactivity for this problem is called 'Building Houses ', but 'Building free ' and also 'Building Houses with Side Views ' are both excellent resources for problems of this kind. As always, Java can take a few seconds to load.

In 'Building free ' you need to type 5 into the size box if you want to match the context of this Perforated Cube problem.

The information above is repeated on the Hint page which also contain this excellent video of one solution for E, S and H - video of an ESH solution