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 napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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.
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.
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 ?
Can you find the area of a parallelogram defined by two vectors?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
What is the same and what is different about these circle
questions? What connections can you make?
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?
There is a particular value of x, and a value of y to go with it,
which make all five expressions equal in value, can you find that
x, y pair ?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
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
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?
Which of these games would you play to give yourself the best possible chance of winning a prize?
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 ?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you maximise the area available to a grazing goat?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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. . . .
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
How many different symmetrical shapes can you make by shading triangles or squares?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
Explore the effect of reflecting in two parallel mirror lines.
Why does this fold create an angle of sixty degrees?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Explore the effect of combining enlargements.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
A jigsaw where pieces only go together if the fractions are
According to an old Indian myth, Sissa ben Dahir was a courtier for
a king. The king decided to reward Sissa for his dedication and
Sissa asked for one grain of rice to be put on the first square. . . .
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
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.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight