Water freezes at 0°Celsius (32°Fahrenheit) and boils at
100°C (212°Fahrenheit). Is there a temperature at which
Celsius and Fahrenheit readings are the same?
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
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?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
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 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 ?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
Can you describe this route to infinity? Where will the arrows take you next?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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.
What is the smallest number with exactly 14 divisors?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
How many different symmetrical shapes can you make by shading triangles or squares?
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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. . . .
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 an efficient method to work out how many handshakes
there would be if hundreds of people met?
Investigate how you can work out what day of the week your birthday
will be on next year, and the year after...
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 circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
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. . . .
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
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?
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?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
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?