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

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

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

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?

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

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

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?

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

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

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

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

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?

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

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

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.

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. . . .

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

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?

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

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?

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.

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...

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

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

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?

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

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. . . .

There are lots of different methods to find out what the shapes are worth - how many can you find?

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?

Which of these games would you play to give yourself the best possible chance of winning a prize?

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

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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?

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

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

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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

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?