Upper Secondary - Complete article index

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?

list

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

Age

11 to 16

I wonder if there's a different way to climb 13 steps for each day of the year... Solving this problem might lead you to a famous number sequence.

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?

list

2D and 3D

Age

11 to 16

The 2d and 3d collection of STEM resources

problem

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?

problem

A fine thing?

Age

16 to 18

Challenge level

Second challenge cipher

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

$A journey into \stemNRICH$

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?

problem

A mean calculation

Age

14 to 16

Challenge level

What is the mean of this set of numbers?

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

Age

14 to 18

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 Roman conversion?

Age

16 to 18

Challenge level

First cipher

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!

Age

16 to 18

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?

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

Additional integrals

Age

16 to 18

Challenge level

Given two integrals as a starting point, can you find links to some related integrals?

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 investigations

Age

14 to 18

Challenge level

Some of our more advanced investigations

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

Age

11 to 16

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

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

list

Adventures with numbers

Age

11 to 16

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.