It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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
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?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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. . . .
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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
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?
Experiment with the interactivity of "rolling" regular polygons,
and explore how the different positions of the red dot affects its
speed at each stage.
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
Can you find all the 4-ball shuffles?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
A metal puzzle which led to some mathematical questions.
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
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?
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?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being
visible at any one time. Is it possible to reorganise these cubes
so that by dipping the large cube into a pot of paint three times
you. . . .
Rotate a copy of the trapezium about the centre of the longest side
of the blue triangle to make a square. Find the area of the square
and then derive a formula for the area of the trapezium.
Prove Pythagoras Theorem using enlargements and scale factors.
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
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?
Work out how to light up the single light. What's the rule?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Can you beat the computer in the challenging strategy game?
A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum. . . .
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Two engines, at opposite ends of a single track railway line, set
off towards one another just as a fly, sitting on the front of one
of the engines, sets off flying along the railway line...
Match the cards of the same value.
Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
Can you set the logic gates so that the number of bulbs which are
on is the same as the number of switches which are on?
Could games evolve by natural selection? Take part in this web experiment to find out!
Can you coach your rowing eight to win?