Can you maximise the area available to a grazing goat?
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?
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 find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Explore the effect of reflecting in two parallel mirror lines.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
If you move the tiles around, can you make squares with different coloured edges?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
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?
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.
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?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Can you find rectangles where the value of the area is the same as the value of the perimeter?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you find the area of a parallelogram defined by two vectors?
Can all unit fractions be written as the sum of two unit fractions?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
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?
What is the same and what is different about these circle
questions? What connections can you make?
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
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.
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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. . . .
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?