What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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.

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

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

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

Is there an efficient way to work out how many factors a large number has?

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. . . .

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

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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

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?

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.

Which set of numbers that add to 10 have the largest product?

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?

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

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?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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?

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

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

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

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?

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?

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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?

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?

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?

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

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

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.

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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?

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?

Explore the effect of combining enlargements.

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

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