Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Can you find the area of a parallelogram defined by two vectors?
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?
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?
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?
Can all unit fractions be written as the sum of two unit fractions?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
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?
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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.
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. . . .
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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 describe this route to infinity? Where will the arrows take you next?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Explore the effect of combining enlargements.
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?
Explore the effect of reflecting in two parallel mirror lines.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
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. . . .
How many different symmetrical shapes can you make by shading triangles or squares?
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?
Can you maximise the area available to a grazing goat?
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?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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 ?
There are lots of different methods to find out what the shapes are worth - how many can you find?
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?
If a sum invested gains 10% each year how long before it has doubled its value?