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

### Air Nets

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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

### Triangles in the middle

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

problem

### The Bridges of Konigsberg

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

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.

article

### When the angles of a triangle don't add up to 180 degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

problem

### Vector walk

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

problem

### Vector journeys

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

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?

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### 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?

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### Iff

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

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### 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.

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### Always Perfect

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

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

### Classic cube

The net of a cube is to be cut from a sheet of card 100 cm square.
What is the maximum volume cube that can be made from a single
piece of card?

problem

### Mach Attack

Have you got the Mach knack? Discover the mathematics behind
exceeding the sound barrier.

problem

### Painting by Numbers

How many different colours of paint would be needed to paint these
pictures by numbers?

problem

### Set Square

A triangle PQR, right angled at P, slides on a horizontal floor
with Q and R in contact with perpendicular walls. What is the locus
of P?

problem

### Wrapping Gifts

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

problem

### Stonehenge

Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.

problem

### Trig reps

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?

problem

### Escriptions

For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.

problem

### Middle Man

Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?

problem

### Maximum Scattering

Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?

problem

### Maths Shop Window

Make a functional window display which will both satisfy the manager and make sense to the shoppers

problem

### Circles ad infinitum

A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?

problem

### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

problem

### Hyperbolic thinking

Explore the properties of these two fascinating functions using trigonometry as a guide.

problem

### Five circuits, seven spins

A circular plate rolls inside a rectangular tray making five
circuits and rotating about its centre seven times. Find the
dimensions of the tray.

problem

### Cheese cutting

In this problem we see how many pieces we can cut a cube of cheese
into using a limited number of slices. How many pieces will you be
able to make?

problem

### Ford Circles

Can you find the link between these beautiful circle patterns and Farey Sequences?

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### Sheep in wolf's clothing

Can you work out what simple structures have been dressed up in these advanced mathematical representations?