The complete list our upper secondary student articles
problem
'Tis whole
Age
14 to 18
Challenge level
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
problem
100m sprint
Age
14 to 16
Challenge level
Anna, Bridget and Carol run a race. Can you work out where Carol was when Anna finished?
problem
11x11 square
Age
11 to 16
Challenge level
Here's a neat trick you can do with an 11 by 11 square...
problem
12345
Age
14 to 16
Challenge level
Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?
problem
13 steps
problem
2-digit square
Age
14 to 16
Challenge level
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?
problem
2009 challenge
Age
11 to 18
Challenge level
We asked what was the most interesting fact that you can find out
about the number 2009. See the solutions that were submitted.
problem
2011 digits
Age
14 to 16
Challenge level
Weekly Problem 10 - 2014
What is the sum of the first $2011$ digits when $20 \div 11$ is written as a decimal?
What is the sum of the first $2011$ digits when $20 \div 11$ is written as a decimal?
problem
2D-3D
Age
16 to 18
Challenge level
Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?
problem
3-4-5 circle
Age
14 to 16
Challenge level
Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?
problem
3-sided
Age
14 to 16
Challenge level
How many triangles can Jack draw with two sides of lengths $6$cm and $8$cm and an area of $7$cm$^2$?
problem
30-60-90 polypuzzle
Age
16 to 18
Challenge level
Re-arrange the pieces of the puzzle to form a rectangle and then to
form an equilateral triangle. Calculate the angles and lengths.
problem
396
Age
14 to 16
Challenge level
The four digits 5, 6, 7 and 8 are put at random in the spaces of
the number : 3 _ 1 _ 4 _ 0 _ 9 2 Calculate the probability that the
answer will be a multiple of 396.
problem
3D treasure hunt
Age
14 to 18
Challenge level
Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?
problem
5 by 5 Mathdokus
Age
7 to 16
Challenge level
Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
problem
6 by 6 Mathdokus
Age
7 to 16
Challenge level
Can you use the clues to complete these 6 by 6 Mathematical Sudokus?
problem
8 methods for three by one
Age
14 to 18
Challenge level
This problem in geometry has been solved in no less than EIGHT ways
by a pair of students. How would you solve it? How many of their
solutions can you follow? How are they the same or different? Which
do you like best?
problem
A big power
Age
16 to 18
Challenge level
Have you ever tried to work out the largest number that your calculator can cope with? What about your computer? Perhaps you tried using powers to make really large numbers. In this problem you will think about how much you can do to understand such numbers when your calculator is less than helpful!
problem
A biggy
Age
14 to 16
Challenge level
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
power.
problem
A brief introduction to complex numbers
Age
14 to 18
Challenge level
In this problem, we define complex numbers and invite you to explore what happens when you add and multiply them.
problem
A brief introduction to the Argand diagram
Age
14 to 18
Challenge level
Complex numbers can be represented graphically using an Argand diagram. This problem explains more...
problem
A change in code
Age
14 to 16
Challenge level
Can you find the percentage increase if the results appear in code?
problem
A circuit problem
Age
16 to 18
Challenge level
Explore the voltages and currents in this interesting circuit configuration.
problem
A close match
Age
16 to 18
Challenge level
Can you massage the parameters of these curves to make them match as closely as possible?
article
A computer program to find magic squares
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
article
A curious collection of bridges
Read about the problem that tickled Euler's curiosity and led to a new branch of mathematics!
problem
A different differential equation
Age
16 to 18
Challenge level
Explore the properties of this different sort of differential
equation.
problem
A drink of water
Age
14 to 16
Challenge level
Weekly Problem 43 - 2015
Rachel and Ross share a bottle of water. Can you work out how much water Rachel drinks?
Rachel and Ross share a bottle of water. Can you work out how much water Rachel drinks?
problem
A frosty puddle
Age
16 to 18
Challenge level
Can you draw a sketch of how Frosty's volume changes as his height decreases?
problem
A function of gradient
Age
16 to 18
Challenge level
Find curves which have gradients of +1 or -1 at various points
article
A journey into stemNRICH
Follow the mathematical journey of a sixth-former as she spent four weeks working on stemNRICH problems.
article
A knight's journey
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
problem
A KS5 proof collection
Age
16 to 18
Challenge level
Here are a collection of statements to prove, to help you to practise writing out clear mathematical proofs.
problem
A little light thinking
Age
14 to 16
Challenge level
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
problem
A long time at the till
Age
14 to 18
Challenge level
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
article
A method of defining coefficients in the equations of chemical reactions
A simple method of defining the coefficients in the equations of chemical reactions with the help of a system of linear algebraic equations.
problem
A population survey
Age
14 to 18
Challenge level
A geographical survey: answer the tiny questionnaire and then analyse all the collected responses...
problem
A powerful matrix
Age
14 to 18
Challenge level
What happens when you find the powers of this matrix?
article
A probability conundrum
What do we mean by probability? This simple problem may challenge your ideas...
problem
A problem of time
Age
14 to 16
Challenge level
Consider a watch face which has identical hands and identical marks
for the hours. It is opposite to a mirror. When is the time as read
direct and in the mirror exactly the same between 6 and 7?
list
A quartet of tetrahedra
In this feature we are exploring and proving some results about tetrahedrons.
problem
A rational search
Age
16 to 18
Challenge level
Investigate constructible images which contain rational areas.
problem
A roll of patterned paper
Age
14 to 16
Challenge level
A design is repeated endlessly along a line - rather like a stream
of paper coming off a roll. Make a strip that matches itself after
rotation, or after reflection
article
A rolling disc - periodic motion
Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?
problem
A sameness surely
Age
14 to 16
Challenge level
Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST
and PU are perpendicular to AB produced. Show that ST + PU = AB
problem
A scale for the solar system
Age
14 to 16
Challenge level
The Earth is further from the Sun than Venus, but how much further?
Twice as far? Ten times?
problem
A shade crossed
Age
14 to 16
Challenge level
Find the area of the shaded region created by the two overlapping
triangles in terms of a and b?
article
A story about absolutely nothing
This article for the young and old talks about the origins of our number system and the important role zero has to play in it.
problem
A swiss sum
Age
16 to 18
Challenge level
Can you use the given image to say something about the sum of an infinite series?
problem
A tangent is ...
Age
16 to 18
Challenge level
What do we REALLY mean when we talk about a tangent to a curve?
problem
A third of the area
Age
14 to 16
Challenge level
The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?
problem
A tilted square
Age
14 to 16
Challenge level
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
problem
A very shiny nose?
Age
16 to 18
Challenge level
This problem explores the biology behind Rudolph's glowing red nose, and introduces the real life phenomena of bacterial quorum sensing.
problem
A well-stirred sample
Age
16 to 18
Challenge level
Typical survey sample sizes are about 1000 people. Why is this?
problem
Ab surd ity
Age
16 to 18
Challenge level
Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of
cuberoot(2+sqrt5)+cuberoot(2-sqrt5).
article
About Pythagorean golden means
What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?
problem
Absolutely!
What can you say about this graph? A number of questions have been suggested to help you look at the graph in different ways. Use these to help you make sense of this and similar graphs.
problem
Absurdity again
Age
16 to 18
Challenge level
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
article
AC/DC circuits
This article, including exercises, gives a thorough grounding in the topic of AC/DC circuits.
problem
Acceptance rate
Age
14 to 16
Challenge level
The mean number of students has changed, so how many students applied to a school?
problem
Added power
Age
14 to 16
Challenge level
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?
list
Addicted to addition
Age
11 to 16
Challenge level
Who would have thought that addition could be so intriguing?
problem
Adding a square to a cube
Age
14 to 16
Challenge level
If you take a number and add its square to its cube, how often will you get a perfect square?
problem
Adding machine
Age
16 to 18
Challenge level
Can you set the logic gates so that this machine can decide how many bulbs have been switched on?
problem
Adding odd numbers
Age
11 to 16
Challenge level
Is there a quick and easy way to calculate the sum of the first 100 odd numbers?
problem
Adding odd numbers (part 2)
Age
16 to 18
Challenge level
Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?
problem
Adding to 400
Age
14 to 16
Challenge level
Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.
problem
Adjacent additions
Age
14 to 16
Challenge level
In a 7-digit numerical code, each group of four consecutive digits adds to 16, and each group of five consecutive digits adds to 19. What is the sum of all 7 digits?
problem
Adjust pendulum
Age
16 to 18
Challenge level
Think about ways to modify the period of a pendulum
list
Advanced mathematical problem solving resources
Age
16 to 18
Challenge level
Resources for students preparing for advanced problem solving examinations.
problem
Advanced mathematics on dotty grids
Age
14 to 18
Challenge level
A dotty grid is a very simple mathematical structure that offers potential for very deep thought...
problem
Advent calendar 2020 - secondary
Our Secondary advent calendar contained twenty-four tasks for the run-up to Christmas, each one from a different past feature.
list
Adventures with complex numbers
Age
14 to 18
Challenge level
This collection is designed to give an introductory taste of complex numbers.
list
Adventures with numbers
This feature introduces you to number theory, a rich source of interesting problems which can lead to surprising results.
problem
Agile algebra
Age
16 to 18
Challenge level
Observe symmetries and engage the power of substitution to solve complicated equations.
problem
Aim high
Age
16 to 18
Challenge level
How do you choose your planting levels to minimise the total loss
at harvest time?
problem
Air nets
Age
7 to 18
Challenge level
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
problem
Air routes
Age
16 to 18
Challenge level
Find the distance of the shortest air route at an altitude of 6000
metres between London and Cape Town given the latitudes and
longitudes. A simple application of scalar products of vectors.
problem
Alberta's age
Age
14 to 16
Challenge level
Alberta won't reveal her age. Can you work it out from these clues?
problem
Algebraic average
Age
14 to 16
Challenge level
The mean of three numbers x, y and z is x. What is the mean of y and z?
problem
Algebraic differences
Age
14 to 16
Challenge level
If $6x−y=21$ and $6y−x=14$, what is the value of $x−y$?
list
Algorithms
Age
16 to 18
Challenge level
Resources to support the teaching of Algorithms, part of the Stage 5 Decision Mathematics collection
problem
Alien currency
Age
14 to 16
Challenge level
The planet Zog has both green and blue bank notes. Can you work out how many zogs two green and three blue bank notes are worth?
problem
All about ratios
Age
16 to 18
Challenge level
A new problem posed by Lyndon Baker who has devised many NRICH
problems over the years.
list
All secondary factors and multiples
Age
11 to 16
Challenge level
Here is a list of all the secondary problems connected with Factors and Multiples
problem
All tangled up
Age
14 to 18
Challenge level
Can you tangle yourself up and reach any fraction?
problem
All tied up
Age
14 to 16
Challenge level
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
problem
All-variables Sudoku
Age
11 to 18
Challenge level
The challenge is to find the values of the variables if you are to solve this Sudoku.