Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
What is the smallest number with exactly 14 divisors?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Find the sum of the series.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
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.
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?
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
Which set of numbers that add to 10 have the largest product?
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?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
In 15 years' time my age will be the square of my age 15 years ago.
Can you work out my age, and when I had other special birthdays?
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...
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?
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 number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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. . . .
Why does this fold create an angle of sixty degrees?
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 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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
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 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?
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?
Explore the effect of combining enlargements.
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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
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?