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?

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?

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

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?

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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 car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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.

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.

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

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. . . .

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

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

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?

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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?

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.

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

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?

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

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

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

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

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

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?

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

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.

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?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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.