Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Which set of numbers that add to 10 have the largest product?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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.
If a sum invested gains 10% each year how long before it has
doubled its value?
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. . . .
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 ?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
There are lots of different methods to find out what the shapes are worth - how many can you find?
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 an efficient method to work out how many handshakes
there would be if hundreds of people met?
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 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Substitute -1, -2 or -3, into an algebraic expression and you'll
get three results. Is it possible to tell in advance which of those
three will be the largest ?
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?
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?
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. . . .
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...
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
Can you work out the dimensions of the three cubes?
Explore the effect of combining enlargements.
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you find the area of a parallelogram defined by two vectors?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Can all unit fractions be written as the sum of two unit fractions?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
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.
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?