What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Find the sum of the series.
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 largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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?
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?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
If a sum invested gains 10% each year how long before it has
doubled its value?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
What is the smallest number with exactly 14 divisors?
Find the decimal equivalents of the fractions one ninth, one ninety
ninth, one nine hundred and ninety ninth etc. Explain the pattern
you get and generalise.
The clues for this Sudoku are the product of the numbers in adjacent squares.
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
Which set of numbers that add to 10 have the largest product?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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
There are lots of different methods to find out what the shapes are worth - how many can you find?
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. . . .
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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 ?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 ?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you find the area of a parallelogram defined by two vectors?
What is the same and what is different about these circle
questions? What connections can you make?
Explore the effect of combining enlargements.
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?
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 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?
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 size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Some people offer advice on how to win at games of chance, or how
to influence probability in your favour. Can you decide whether
advice is good or not?
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 ?
Can you maximise the area available to a grazing goat?
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. . . .
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?
Why does this fold create an angle of sixty degrees?