NRICH has always had good solutions from Madras College in St
Andrew's, Scotland but the solutions to this problem were truly
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
What is the same and what is different about these circle
questions? What connections can you make?
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?
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?
There are lots of different methods to find out what the shapes are worth - how many can you find?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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. . . .
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Find the sum of the series.
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?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Why does this fold create an angle of sixty degrees?
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?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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 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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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 ?
Stick some cubes together to make a cuboid. Find two of the angles
by as many different methods as you can devise.
Can you describe this route to infinity? Where will the arrows take you next?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
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. . . .
Which of these games would you play to give yourself the best possible chance of winning a prize?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
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. . . .
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
If a sum invested gains 10% each year how long before it has
doubled its value?
Your school has been left a million pounds in the will of an ex-
pupil. What model of investment and spending would you use in order
to ensure the best return on the money?
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?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
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
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Use the differences to find the solution to this Sudoku.
The clues for this Sudoku are the product of the numbers in
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.
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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 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?
Investigate how you can work out what day of the week your birthday
will be on next year, and the year after...
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?