A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Why does this fold create an angle of sixty degrees?
What is the same and what is different about these circle
questions? What connections can you make?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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?
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. . . .
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?
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?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
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?
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 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
There are lots of different methods to find out what the shapes are worth - how many can you find?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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. . . .
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. . . .
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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 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?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
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.
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 you find rectangles where the value of the area is the same as the value of the perimeter?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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?
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. . . .
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
What is the smallest number with exactly 14 divisors?
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. . . .
How many different symmetrical shapes can you make by shading triangles or squares?
Use the differences to find the solution to this Sudoku.
In a three-dimensional version of noughts and crosses, how many winning lines 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?
Find the decimal equivalents of the fractions one ninth, one ninety
ninth, one nine hundred and ninety ninth etc. Explain the pattern
you get and generalise.
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?
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. . . .
The clues for this Sudoku are the product of the numbers in adjacent squares.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Can you maximise the area available to a grazing goat?
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.