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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Can you describe this route to infinity? Where will the arrows take you next?
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 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?
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?
Can you work out the dimensions of the three cubes?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What is the same and what is different about these circle
questions? What connections can you make?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle 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
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
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?
Can you maximise the area available to a grazing goat?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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. . . .
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Use the differences to find the solution to this Sudoku.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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.
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?
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.
How many different symmetrical shapes can you make by shading triangles or squares?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Explore the effect of combining enlargements.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Explore the effect of reflecting in two parallel mirror lines.
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Can all unit fractions be written as the sum of two unit fractions?
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 is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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 ?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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.