Can you find rectangles where the value of the area is the same as the value of the perimeter?
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. . . .
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?
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
Can you maximise the area available to a grazing goat?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Why does this fold create an angle of sixty degrees?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Explore the effect of combining enlargements.
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
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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 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?
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.
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?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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.
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
Can all unit fractions be written as the sum of two unit fractions?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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.
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Chris and Jo put two red and four blue ribbons in a box. They each
pick a ribbon from the box without looking. Jo wins if the two
ribbons are the same colour. Is the game fair?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
If you are given the mean, median and mode of five positive whole numbers, can you find the 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?
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?