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.

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

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?

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

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

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

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

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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?

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?

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?

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?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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

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

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

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.

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

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

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

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?

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

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. . . .

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

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.

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?

Use the differences to find the solution to this Sudoku.

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

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?

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

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?

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

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

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

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?

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

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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

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