The complete list our upper secondary student articles

problem

### ' Tis Whole

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

Anna, Bridget and Carol run a race. Can you work out where Carol was when Anna finished?

problem

### 12345

Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?

problem

### 13 Steps

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Favourite

### 2-Digit Square

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

Favourite

### 2009 challenge

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

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

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-sided

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

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

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

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

problem

Favourite

### 5 by 5 Mathdokus

Can you use the clues to complete these 5 by 5 Mathematical Sudokus?

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Favourite

### 8 Methods for Three By One

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

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

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

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

Complex numbers can be represented graphically using an Argand diagram. This problem explains more...

problem

Favourite

### A Change in Code

Can you find the percentage increase if the results appear in code?

problem

### A circuit problem

Explore the voltages and currents in this interesting circuit configuration.

problem

### A close match

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

Explore the properties of this different sort of differential
equation.

problem

### A Drink of Water

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

Can you draw a sketch of how Frosty's volume changes as his height decreases?

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

Here are a collection of statements to prove, to help you to practise writing out clear mathematical proofs.

problem

Favourite

### A little light thinking

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

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

A geographical survey: answer the tiny questionnaire and then analyse all the collected responses...

article

### A probability conundrum

What do we mean by probability? This simple problem may challenge your ideas...

problem

### A Problem of time

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 Roll Of Patterned Paper

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

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

The Earth is further from the Sun than Venus, but how much further?
Twice as far? Ten times?

problem

### A Shade Crossed

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

Can you use the given image to say something about the sum of an infinite series?

problem

### A third of the area

The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?

problem

### A Tilted Square

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?

This problem explores the biology behind Rudolph's glowing red nose, and introduces the real life phenomena of bacterial quorum sensing.

problem

Favourite

### Ab Surd Ity

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

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

The mean number of students has changed, so how many students applied to a school?

list

### Addicted to Addition

problem

### Adding a square to a cube

If you take a number and add its square to its cube, how often will you get a perfect square?

problem

### Adding machine

Can you set the logic gates so that this machine can decide how many bulbs have been switched on?

problem

### Adding odd numbers

Is there a quick and easy way to calculate the sum of the first 100 odd numbers?

problem

### Adding odd numbers (part 2)

Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?

problem

### Adding to 400

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

### Additional integrals

problem

### Adjacent Additions

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?

list

### Advanced Mathematical Problem Solving Resources

Resources for students preparing for advanced problem solving examinations.

problem

### Advanced Mathematics on Dotty Grids

A dotty grid is a very simple mathematical structure that offers potential for very deep thought...

problem

### Advent Calendar 2011 - Secondary

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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Favourite

### 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

This collection is designed to give an introductory taste of complex numbers.

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### 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

Observe symmetries and engage the power of substitution to solve complicated equations.

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Favourite

### Aim high

How do you choose your planting levels to minimise the total loss
at harvest time?

problem

### Air Nets

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

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.

list

### Algorithms

Resources to support the teaching of Algorithms, part of the Stage 5 Decision Mathematics collection

problem

### Alien Currency

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?

list

### All about averages

problem

Favourite

### All About Ratios

A new problem posed by Lyndon Baker who has devised many NRICH
problems over the years.

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### All Secondary Factors and Multiples

Here is a list of all the secondary problems connected with Factors and Multiples