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

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

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

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

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

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

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

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?

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?

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...

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

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

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

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

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?

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?

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?

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.

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.

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

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?

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?

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.

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

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

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

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. . . .

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.

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

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?

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

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

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 ?

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?

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

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

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

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?

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?

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

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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

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 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?