Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

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?

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

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

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

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?

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?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

Can you maximise the area available to a grazing goat?

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

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

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

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

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?

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?

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?

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?

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

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.

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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 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.

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?

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...

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

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

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.

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

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.

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?

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?

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

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?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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

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?

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

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

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?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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