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?
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.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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. . . .
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?
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.
A jigsaw where pieces only go together if the fractions are
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
Can all unit fractions be written as the sum of two unit fractions?
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
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?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Which of these games would you play to give yourself the best possible chance of winning a prize?
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
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 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. . . .
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
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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. . . .
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. . . .
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.
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.
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
What is the same and what is different about these circle
questions? What connections can you make?
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
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?
Substitute -1, -2 or -3, into an algebraic expression and you'll
get three results. Is it possible to tell in advance which of those
three will be the largest ?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
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.
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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 ?