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?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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

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

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

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

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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

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

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.

Explore the effect of combining enlargements.

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

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

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.

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

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

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?

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.

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?

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?

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?

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

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?

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

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?

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.

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

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

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

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 ?

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

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

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?

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?

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

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

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?

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

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?

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?

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

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?

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

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.