Visualising and representing

problem
Instant insanity

Age

11 to 18

Challenge level

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
problem
Three cubes

Age

14 to 16

Challenge level

Can you work out the dimensions of the three cubes?
problem
Orthogonal circle

Age

16 to 18

Challenge level

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
problem
Platonic planet

Age

14 to 16

Challenge level

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?
problem
Stonehenge

Age

16 to 18

Challenge level

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
problem
Natural sum

Age

14 to 16

Challenge level

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.
problem
Proximity

Age

14 to 16

Challenge level

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
problem
Picture story

Age

14 to 16

Challenge level

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
problem
Contact

Age

14 to 16

Challenge level

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?
problem
Bendy quad

Age

14 to 16

Challenge level

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.