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 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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 is the same and what is different about these circle
questions? What connections can you make?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Can you maximise the area available to a grazing goat?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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.
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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
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 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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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. . . .
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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 it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
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.
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 hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
What is the smallest number with exactly 14 divisors?
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
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 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?
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 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. . . .
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Which of these games would you play to give yourself the best possible chance of winning a prize?
A jigsaw where pieces only go together if the fractions are
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.