Or search by topic
Which armies can be arranged in hollow square fighting formations?
There are unexpected discoveries to be made about square numbers...
What is special about the difference between squares of numbers adjacent to multiples of three?
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
Charlie has moved between countries and the average income of both has increased. How can this be so?
What's the largest volume of box you can make from a square of paper?
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
It would be nice to have a strategy for disentangling any tangled ropes...
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
A game in which players take it in turns to choose a number. Can you block your opponent?
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 the area of a parallelogram defined by two vectors?
Can you find an efficent way to mix paints in any ratio?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Can you find the values at the vertices when you know the values on the edges?
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
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?
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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 loser is the player who takes the last counter.
Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?