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?
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?
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?
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. . . .
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. . . .
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?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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?
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
Can you maximise the area available to a grazing goat?
Can you describe this route to infinity? Where will the arrows take you next?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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. . . .
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
How many solutions can you find to this sum? Each of the different letters stands for a different number.
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
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. . . .
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Use the differences to find the solution to this Sudoku.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
If you are given the mean, median and mode of five positive whole
numbers, can you find the numbers?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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. . . .
What is the same and what is different about these circle
questions? What connections can you make?
How many different symmetrical shapes can you make by shading triangles or squares?
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...
The clues for this Sudoku are the product of the numbers in adjacent squares.
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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?
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
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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 ?