Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
Your school has been left a million pounds in the will of an ex-
pupil. What model of investment and spending would you use in order
to ensure the best return on the money?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Which set of numbers that add to 10 have the largest product?
Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?
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?
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
A spider is sitting in the middle of one of the smallest walls in a
room and a fly is resting beside the window. What is the shortest
distance the spider would have to crawl to catch the fly?
Here's a chance to work with large numbers...
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
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. . . .
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
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?
Is there an efficient way to work out how many factors a large number has?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
Can you work out the dimensions of the three cubes?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
There are lots of different methods to find out what the shapes are worth - how many can you find?
What is the same and what is different about these circle
questions? What connections can you make?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Which of these games would you play to give yourself the best possible chance of winning a prize?
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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72