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?
Is there an efficient way to work out how many factors a large number has?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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 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.
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
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?
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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 ?
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 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?
Explore the effect of combining enlargements.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
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?
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?
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?
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 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?
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. . . .
Explore the effect of reflecting in two parallel mirror lines.
Can you describe this route to infinity? Where will the arrows take you next?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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. . . .