Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
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 you move the tiles around, can you make squares with different coloured edges?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .
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?
Can you describe this route to infinity? Where will the arrows take you next?
Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?
If a sum invested gains 10% each year how long before it has doubled its value?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Explore the effect of reflecting in two parallel mirror lines.
How many different symmetrical shapes can you make by shading triangles or squares?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?
Can you maximise the area available to a grazing goat?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
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?
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
Can you find the area of a parallelogram defined by two vectors?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
What is the same and what is different about these circle questions? What connections can you make?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Which of these games would you play to give yourself the best possible chance of winning a prize?