Visualising - Advanced

game

Alquerque

This game for two, was played in ancient Egypt as far back as 1400 BC. The game was taken by the Moors to Spain, where it is mentioned in 13th century manuscripts, and the Spanish name Alquerque derives from the Arabic El- quirkat. Watch out for being 'huffed'.

game

Pumpkin patch

A game for two players based on a game from the Somali people of Africa. The first player to pick all the other's 'pumpkins' is the winner.

game

Seega

An ancient game for two from Egypt. You'll need twelve distinctive 'stones' each to play. You could chalk out the board on the ground - do ask permission first.

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

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.

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

Age

11 to 18

Challenge level

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

problem

The bridges of Konigsberg

Age

11 to 18

Challenge level

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

game

Yih or Luk tsut k'i or Three Men's Morris

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and knot arithmetic.

problem

Instant insanity

Age

11 to 18

Challenge level

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

Proofs with pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

Curve hunter

Age

14 to 18

Challenge level

This problem challenges you to sketch curves with different properties.

problem

Favourite

Vector journeys

Age

14 to 18

Challenge level

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

Favourite

Summing geometric progressions

Age

14 to 18

Challenge level

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

problem

Can you traverse it?

Age

14 to 18

Challenge level

How can you decide if a graph is traversable?

problem

Favourite

Vector walk

Age

14 to 18

Challenge level

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

problem

Bent out of shape

Age

14 to 18

Challenge level

An introduction to bond angle geometry.

problem

Favourite

What's that graph?

Age

14 to 18

Challenge level

Can you work out which processes are represented by the graphs?

problem

Favourite

Iff

Age

14 to 18

Challenge level

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

problem

Tetra square

Age

14 to 18

Challenge level

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

Vanishing point

Age

14 to 18

Challenge level

How can visual patterns be used to prove sums of series?

problem

Favourite

The root of the problem

Age

14 to 18

Challenge level

Find the sum of this series of surds.

problem

Favourite

Always perfect

Age

14 to 18

Challenge level

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

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

Favourite

Back fitter

Age

14 to 18

Challenge level

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?

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

Mach attack

Age

16 to 18

Challenge level

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

problem

Painting by numbers

Age

16 to 18

Challenge level

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

problem

Painting by functions

Age

16 to 18

Challenge level

Use functions to create minimalist versions of works of art.

problem

Set square

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

problem

Escriptions

Age

16 to 18

Challenge level

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

problem

Middle man

Age

16 to 18

Challenge level

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

Polar bearings

Age

16 to 18

Challenge level

What on earth are polar coordinates, and why would you want to use them?

problem

Favourite

Parabella

Age

16 to 18

Challenge level

This is a beautiful result involving a parabola and parallels.

problem

Trig reps

Age

16 to 18

Challenge level

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

problem

Classic cube

Age

16 to 18

Challenge level

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

Fitting flat shapes

Age

16 to 18

Challenge level

How efficiently can various flat shapes be fitted together?

problem

Whose line graph is it anyway?

Age

16 to 18

Challenge level

Which line graph, equations and physical processes go together?

problem

Coordinated crystals

Age

16 to 18

Challenge level

Explore the lattice and vector structure of this crystal.

problem

Classical means

Age

16 to 18

Challenge level

Use the diagram to investigate the classical Pythagorean means.

problem

Maths shop window

Age

16 to 18

Challenge level

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

problem

Favourite

Circles ad infinitum

Age

16 to 18

Challenge level

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?