How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

Use the isometric grid paper to find the different polygons.

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

Investigate these hexagons drawn from different sized equilateral triangles.

Look at the mathematics that is all around us - this circular window is a wonderful example.

What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

This LOGO challenge starts by looking at 10-sided polygons then generalises the findings to any polygon, putting particular emphasis on external angles

An environment that enables you to investigate tessellations of regular polygons

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Thinking of circles as polygons with an infinite number of sides - but how does this help us with our understanding of the circumference of circle as pi x d? This challenge investigates. . . .

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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?

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

Can you reproduce the design comprising a series of concentric circles? Test your understanding of the realtionship betwwn the circumference and diameter of a circle.

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring?

See if you can anticipate successive 'generations' of the two animals shown here.

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.

Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle?

What shape and size of drinks mat is best for flipping and catching?

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

What fractions of the largest circle are the two shaded regions?

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

A metal puzzle which led to some mathematical questions.

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.