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?
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 to. . . .
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you maximise the area available to a grazing goat?
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.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Can all unit fractions be written as the sum of two unit fractions?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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. . . .
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?
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?
Explore the effect of reflecting in two parallel mirror lines.
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?
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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 combining enlargements.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you find the area of a parallelogram defined by two vectors?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
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?
What is the smallest number with exactly 14 divisors?
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 is the same and what is different about these circle
questions? What connections can you make?
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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
Can you describe this route to infinity? Where will the arrows take you next?
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 Egyptians expressed all fractions as the sum of different unit
fractions. The Greedy Algorithm might provide us with an efficient
way of doing this.
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...