Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you find the area of a parallelogram defined by two vectors?
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 maximise the area available to a grazing goat?
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?
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. . . .
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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
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?
Powers of numbers behave in surprising ways. Take a look at some of
these and try to explain why they are true.
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What is the smallest number with exactly 14 divisors?
What is the same and what is different about these circle
questions? What connections can you make?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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. . . .
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. . . .
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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?
Different combinations of the weights available allow you to make different totals. Which totals 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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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?
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.
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. . . .
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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.
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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 jigsaw where pieces only go together if the fractions are
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
The clues for this Sudoku are the product of the numbers in
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
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 number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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. . . .
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.
Here's a chance to work with large numbers...