Can you maximise the area available to a grazing goat?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
How many winning lines can you make in a three-dimensional version of noughts and 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?
Can you describe this route to infinity? Where will the arrows take you next?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
If you move the tiles around, can you make squares with different coloured edges?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
Explore the effect of reflecting in two parallel mirror lines.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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.
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?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can all unit fractions be written as the sum of two unit fractions?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Explore the effect of combining enlargements.
How many different symmetrical shapes can you make by shading triangles or squares?
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?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
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.
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. . . .
What is the same and what is different about these circle questions? What connections can you make?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
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?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Can you find the area of a parallelogram defined by two vectors?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
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?