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

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

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

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

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

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.

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

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

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

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

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?

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?

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

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.

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?

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...

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

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

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

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?

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

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

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

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?

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

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.

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.

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 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.

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

Use the differences to find the solution to this Sudoku.

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

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?

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. . . .

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?

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

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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

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?

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. . . .

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.

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

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

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

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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?

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?