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
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Can you find all the 4-ball shuffles?
Six balls of various colours are randomly shaken into a trianglular
arrangement. What is the probability of having at least one red in
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.
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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
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. . . .
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.
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
7 balls are shaken in a container. You win if the two blue balls
touch. What is the probability of winning?
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 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. . . .
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
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
Can you discover whether this is a fair game?
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
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?
Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?
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
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 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. . . .
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
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?
Meg and Mo need to hang their marbles so that they balance. Use the
interactivity to experiment and find out what they need to do.
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.
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
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?
Can you make a right-angled triangle on this peg-board by joining
up three points round the edge?
Practise your diamond mining skills and your x,y coordination in this homage to Pacman.
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?
A train building game for 2 players.
Can you locate the lost giraffe? Input coordinates to help you
search and find the giraffe in the fewest guesses.
Match pairs of cards so that they have equivalent ratios.
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.
An animation that helps you understand the game of Nim.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Find out what a "fault-free" rectangle is and try to make some of
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?
A generic circular pegboard resource.