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?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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?
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?
Which set of numbers that add to 10 have the largest product?
Can all unit fractions be written as the sum of two unit fractions?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Can you maximise the area available to a grazing goat?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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. . . .
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
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?
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?
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?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
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. . . .
In a three-dimensional version of noughts and crosses, how many winning lines 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 ?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you describe this route to infinity? Where will the arrows take you next?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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?