interactivity

### Introducing NRICH TWILGO

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

problem

### Prime Magic

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

problem

### Like a Circle in a Spiral

A cheap and simple toy with lots of mathematics. Can you interpret
the images that are produced? Can you predict the pattern that will
be produced using different wheels?

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

### Clocking off

I found these clocks in the Arts Centre at the University of
Warwick intriguing - do they really need four clocks and what times
would be ambiguous with only two or three of them?

problem

### What is the question?

These pictures and answers leave the viewer with the problem "What
is the Question". Can you give the question and how the answer
follows?

problem

### Parallelogram It

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.

problem

### The Bridges of Konigsberg

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

problem

### Semi-regular Tessellations

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

problem

### Rhombus It

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.

problem

### Marbles in a box

How many winning lines can you make in a three-dimensional version of noughts and crosses?

article

### Ding Dong Bell

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

problem

### Searching for mean(ing)

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

problem

### Hamiltonian Cube

Weekly Problem 36 - 2007

Find the length along the shortest path passing through certain points on the cube.

Find the length along the shortest path passing through certain points on the cube.

problem

### LOGO Challenge - Circles as animals

See if you can anticipate successive 'generations' of the two
animals shown here.

article

### Shaping the universe I - planet Earth

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

problem

### Funny Factorisation

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

problem

### LOGO Challenge - Triangles-Squares-Stars

Can you recreate these designs? What are the basic units? What
movement is required between each unit? Some elegant use of
procedures will help - variables not essential.

article

### Shaping the universe II - the solar system

The second in a series of articles on visualising and modelling shapes in the history of astronomy.

article

### Shaping the universe III - to infinity and beyond

The third installment in our series on the shape of astronomical systems, this article explores galaxies and the universe beyond our solar system.

problem

### Charting success

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

problem

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

problem

### Nine Colours

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

problem

### Triangles in the middle

This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.

problem

### Charting more success

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

problem

### Twelve Cubed

A wooden cube with edges of length 12cm is cut into cubes with edges of length 1cm. What is the total length of the all the edges of these centimetre cubes?

problem

### Semicircular Design

Weekly Problem 9 - 2016

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

problem

### Corridors

A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
corner.

problem

### Facial Sums

Can you make the numbers around each face of this solid add up to the same total?

problem

### Just rolling round

P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
of P?

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

### Steel Cables

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

problem

### Doesn't add up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

problem

### Dicey Directions

An ordinary die is placed on a horizontal table with the '1' face facing East... In which direction is the '1' face facing after this sequence of moves?

problem

### Travelling by Train

Stephen stops at Darlington on his way to Durham. At what time does he arrive at Durham?

problem

### Gnomon dimensions

These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
these connections.

problem

### Speeding boats

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

problem

### Always Perfect

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

problem

### Coke machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

problem

### Slippage

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance up the wall which the ladder can reach?

problem

### Fermat's Poser

Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.

problem

### Spotting the loophole

A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
algebra?

problem

### Oldest and Youngest

Edith had 9 children at 15 month intervals. If the oldest is now six times as old as the youngest, how old is her youngest child?

problem

### Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

problem

### Back fitter

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

problem

### Perfectly Square

The sums of the squares of three related numbers is also a perfect square - can you explain why?

problem

### Just Opposite

A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?

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?

problem

### Making Tracks

A bicycle passes along a path and leaves some tracks. Is it
possible to say which track was made by the front wheel and which
by the back wheel?

problem

### Tetra Square

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

problem

### Changing Places

Place a red counter in the top left corner of a 4x4 array, which is
covered by 14 other smaller counters, leaving a gap in the bottom
right hand corner (HOME). What is the smallest number of moves it
will take to move the red counter to HOME?

problem

### Packing Boxes

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

problem

### Triangles within Triangles

Can you find a rule which connects consecutive triangular numbers?

problem

### Hypotenuse Lattice points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

problem

### One Out One Under

Imagine a stack of numbered cards with one on top. Discard the top,
put the next card to the bottom and repeat continuously. Can you
predict the last card?

problem

### Tennis Training

After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?

problem

### Concrete calculation

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

problem

### Surprising Transformations

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

problem

### Painted Cube

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?

problem

### Newspaper Sheets

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

problem

### Triangles within Squares

Can you find a rule which relates triangular numbers to square numbers?

problem

### Triangles within Pentagons

Show that all pentagonal numbers are one third of a triangular number.

problem

### Relative Time

Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

problem

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.

problem

### The perforated cube

A cube is made from smaller cubes, 5 by 5 by 5, then some of those
cubes are removed. Can you make the specified shapes, and what is
the most and least number of cubes required ?

problem

### Vector walk

Starting with two basic vector steps, which destinations can you reach on a vector walk?

problem

### Tetrahedra Tester

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

problem

### Speed-time problems at the Olympics

Have you ever wondered what it would be like to race against Usain Bolt?

problem

### Eulerian

Weekly Problem 37 - 2014

Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

problem

### Penta Colour

In how many different ways can I colour the five edges of a pentagon so that no two adjacent edges are the same colour?

problem

### Negatively Triangular

How many intersections do you expect from four straight lines ? Which three lines enclose a triangle with negative co-ordinates for every point ?

problem

### Tilting Triangles

A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?

problem

### Which is cheaper?

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

problem

### Bendy Quad

Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.

problem

### Pyramidal n-gon

The base of a pyramid has n edges. What is the difference between the number of edges the pyramid has and the number of faces the pyramid has?

problem

### Escalator

At Holborn underground station there is a very long escalator. Two
people are in a hurry and so climb the escalator as it is moving
upwards, thus adding their speed to that of the moving steps. ...
How many steps are there on the escalator?

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

### Summing squares

Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?

problem

### The Spider and the Fly

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?

problem

### Which is bigger?

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

problem

### Contact

A circular plate rolls in contact with the sides of a rectangular
tray. How much of its circumference comes into contact with the
sides of the tray when it rolls around one circuit?

problem

### In or Out?

Weekly Problem 52 - 2014

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

problem

### All Tied Up

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

### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

problem

### Cubic Covering

A blue cube has blue cubes glued on all of its faces. Yellow cubes are then glued onto all the visible blue facces. How many yellow cubes are needed?