A jigsaw where pieces only go together if the fractions are equivalent.

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

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

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 arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

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

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?

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

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

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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

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

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

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?

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

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: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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?

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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".

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?

Can you describe this route to infinity? Where will the arrows take you next?

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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

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?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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.

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Explore the effect of combining enlargements.

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?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?