What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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.
What is the smallest number with exactly 14 divisors?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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.
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?
Which set of numbers that add to 10 have the largest product?
Find the sum of the series.
Can all unit fractions be written as the sum of two unit fractions?
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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
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.
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?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
The clues for this Sudoku are the product of the numbers in adjacent squares.
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
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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.
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 ?
What is the same and what is different about these circle
questions? What connections can you make?
Use the differences to find the solution to this Sudoku.
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 find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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?