Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
What is the smallest number with exactly 14 divisors?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
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.
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Find the sum of the series.
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting 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?
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?
Which set of numbers that add to 10 have the largest product?
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?
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?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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 diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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 ?
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Can you find the area of a parallelogram defined by two vectors?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
Can you maximise the area available to a grazing goat?
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 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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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. . . .
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?