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?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
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?
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 see how to build a harmonic triangle? Can you work out the next two rows?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Find the sum of the series.
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?
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?
Can all unit fractions be written as the sum of two unit fractions?
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?
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?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
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?
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. . . .
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?
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. . . .
Can you find the area of a parallelogram defined by two vectors?
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. . . .
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?
Which of these games would you play to give yourself the best possible chance of winning a prize?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Why does this fold create an angle of sixty degrees?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
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
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?
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. . . .
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?
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 "nodes".
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?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
A plastic funnel is used to pour liquids through narrow apertures.
What shape funnel would use the least amount of plastic to
manufacture for any specific volume ?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?