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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
If a sum invested gains 10% each year how long before it has
doubled its value?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Can you find the area of a parallelogram defined by two vectors?
Can you describe this route to infinity? Where will the arrows take you next?
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 all unit fractions be written as the sum of two unit fractions?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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 ?
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
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?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
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?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
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.
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.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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.
All CD Heaven stores were given the same number of a popular CD to
sell for £24. In their two week sale each store reduces the
price of the CD by 25% ... How many CDs did the store sell at. . . .
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?
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?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
What is the smallest number with exactly 14 divisors?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Find the decimal equivalents of the fractions one ninth, one ninety
ninth, one nine hundred and ninety ninth etc. Explain the pattern
you get and generalise.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Which set of numbers that add to 10 have the largest product?