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?

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

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

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

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

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

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?

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.

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

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...

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

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

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?

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

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

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

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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?

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

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...

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.

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?

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?

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

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...

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?

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. . . .

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?

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 ?

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

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

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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

Explore the effect of combining enlargements.

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

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

If you move the tiles around, can you make squares with different coloured edges?

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

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?

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?