Visualising and representing

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

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.

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

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

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?

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.

Do you have enough information to work out the area of the shaded quadrilateral?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?