Is there an efficient way to work out how many factors a large number has?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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...
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
The clues for this Sudoku are the product of the numbers in adjacent squares.
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
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you find the area of a parallelogram defined by two vectors?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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 ?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
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. . . .