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

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

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?

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

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?

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.

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?

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?

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

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

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.

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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

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 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?

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

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

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

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

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

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.

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?

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?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

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?

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

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?

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. . . .

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?

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.

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?

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

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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

Explore the effect of combining enlargements.

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

Explore the effect of reflecting in two parallel mirror lines.

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