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?

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

Weekly Problem 39 - 2011

Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

Weekly Problem 38 - 2011

Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

Weekly Problem 37 - 2011

Rotating a pencil twice about two different points gives surprising results...

Weekly Problem 36 - 2011

Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.

The three solutions from Rachael and Katy, James and Derek show an interesting range of approaches and detail.

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.