Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
If a sum invested gains 10% each year how long before it has
doubled its value?
Which set of numbers that add to 10 have the largest product?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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 ?
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. . . .
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?
Can you describe this route to infinity? Where will the arrows take you next?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Investigate how you can work out what day of the week your birthday
will be on next year, and the year after...
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
All CD Heaven stores were given the same number of a popular CD to
sell for £24. In their two week sale each store reduces the
price of the CD by 25% ... How many CDs did the store sell at. . . .
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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 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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
There are lots of different methods to find out what the shapes are worth - how many can you find?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
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?
It is known that the area of the largest equilateral triangular
section of a cube is 140sq cm. What is the side length of the cube?
The distances between the centres of two adjacent faces of. . . .
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
A plastic funnel is used to pour liquids through narrow apertures.
What shape funnel would use the least amount of plastic to
manufacture for any specific volume ?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you maximise the area available to a grazing goat?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Explore the effect of combining enlargements.
How many different symmetrical shapes can you make by shading triangles or squares?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Explore the effect of reflecting in two parallel mirror lines.
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?