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

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.

A jigsaw where pieces only go together if the fractions are equivalent.

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

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 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?

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?

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.

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

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

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?

Explore the effect of reflecting in two parallel mirror lines.

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

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

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

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?

Explore the effect of combining enlargements.

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?

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

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?

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

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

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

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?

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

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

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?

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

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?

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.

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?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

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.

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

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?

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

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?

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?

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?

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.

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

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?

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

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.