Challenge Level

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Challenge Level

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Challenge Level

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Challenge Level

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Challenge Level

It would be nice to have a strategy for disentangling any tangled ropes...

Challenge Level

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

Challenge Level

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Challenge Level

How much of the square is coloured blue? How will the pattern continue?

Challenge Level

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Challenge Level

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

Challenge Level

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

Challenge Level

Kyle and his teacher disagree about his test score - who is right?

Challenge Level

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

Challenge Level

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Challenge Level

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Challenge Level

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

Challenge Level

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Challenge Level

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Challenge Level

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Challenge Level

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Challenge Level

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Challenge Level

There are lots of ideas to explore in these sequences of ordered fractions.

Challenge Level

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Challenge Level

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Challenge Level

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Challenge Level

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Challenge Level

Use these four dominoes to make a square that has the same number of dots on each side.

Challenge Level

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

Challenge Level

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Challenge Level

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

Challenge Level

Join pentagons together edge to edge. Will they form a ring?

Use the applet to explore the area of a parallelogram and how it relates to vectors.

Challenge Level

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?

Week 2

How well can you estimate angles? Playing this game could improve your skills.

Use the applet to make some squares. What patterns do you notice in the coordinates?

Challenge Level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Challenge Level

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Challenge Level

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Challenge Level

Can you find triangles on a 9-point circle? Can you work out their angles?

Move the corner of the rectangle. Can you work out what the purple number represents?

Challenge Level

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.

Challenge Level

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?

Challenge Level

Under what circumstances can you rearrange a big square to make three smaller squares?