Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Is there an efficient way to work out how many factors a large number has?
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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.
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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Can you find the area of a parallelogram defined by two vectors?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Which set of numbers that add to 10 have the largest product?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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. . . .
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 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 ?
Five children went into the sweet shop after school. There were
choco bars, chews, mini eggs and lollypops, all costing under 50p.
Suggest a way in which Nathan could spend all his money.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Start with two numbers and generate a sequence where the next number is the mean of the last two 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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Explore the effect of reflecting in two parallel mirror lines.
Can you describe this route to infinity? Where will the arrows take you next?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
Explore the effect of combining enlargements.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?
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?
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. . . .
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Can all unit fractions be written as the sum of two unit fractions?