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.

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?

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

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

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

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

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

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?

Explore the effect of combining enlargements.

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

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

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?

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?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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?

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

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 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?

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

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...

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Can you maximise the area available to a grazing goat?

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

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?

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

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?

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

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.

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?

If a sum invested gains 10% each year how long before it has doubled its value?

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

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

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Use the differences to find the solution to this Sudoku.

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

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

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

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

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?