### 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...

### 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 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