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A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Does weight confer an advantage to shot putters?

What's the largest volume of box you can make from a square of paper?

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?