Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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 describe this route to infinity? Where will the arrows take you next?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
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 ?
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?
Explore the effect of reflecting in two parallel mirror lines.
How many different symmetrical shapes can you make by shading triangles or squares?
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?
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?
If a sum invested gains 10% each year how long before it has doubled its value?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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...
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?
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?
Can you find the area of a parallelogram defined by two vectors?
All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".
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 maximise the area available to a grazing goat?
Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?
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?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try. . . .
Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...
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...
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
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.
Explore the effect of combining enlargements.
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?