Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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.

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?

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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 ?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

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?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

If a sum invested gains 10% each year how long before it has doubled its value?

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?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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.

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 same and what is different about these circle questions? What connections can you make?

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?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Can all unit fractions be written as the sum of two unit fractions?

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

Explore the effect of reflecting in two parallel mirror lines.

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?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Explore the effect of combining enlargements.

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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.

How many different symmetrical shapes can you make by shading triangles or squares?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Can you find the area of a parallelogram defined by two vectors?

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?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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 ?

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?