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?
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?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Which set of numbers that add to 10 have the largest product?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
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.
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...
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Find the sum of the series.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
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?
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?
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. . . .
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 ?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Explore the effect of reflecting in two parallel mirror lines.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you maximise the area available to a grazing goat?