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. . . .
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
What is the same and what is different about these circle
questions? What connections can you make?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A decorator can buy pink paint from two manufacturers. What is the
least number he would need of each type in order to produce
different shades of pink.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Is it always possible to combine two paints made up in the ratios
1:x and 1:y and turn them into paint made up in the ratio a:b ? Can
you find an efficent way of doing this?
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?
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?
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 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
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Think of two whole numbers under 10. Take one of them and add 1.
Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your
second number. Add 2. Double again. Subtract 8. Halve this. . . .
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
It is known that the area of the largest equilateral triangular
section of a cube is 140sq cm. What is the side length of the cube?
The distances between the centres of two adjacent faces of. . . .
Explore the effect of combining enlargements.
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?
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
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.
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
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?
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?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
The Egyptians expressed all fractions as the sum of different unit
fractions. The Greedy Algorithm might provide us with an efficient
way of doing this.
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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. . . .
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
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?
What is the smallest number with exactly 14 divisors?
A jigsaw where pieces only go together if the fractions are
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.
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?
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.