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

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

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

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

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?

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

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

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?

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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.

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?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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

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?

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

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?

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?

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?

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.

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

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

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?

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?

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

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

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. . . .

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. . . .

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?

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 ?

Explore the effect of combining enlargements.

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

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

Explore the effect of reflecting in two parallel mirror lines.

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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.

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

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

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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?