List

Visualising - Advanced

Pumpkin Patch
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.
Seega
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.
Alquerque
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'.
Introducing NRICH TWILGO
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?
Air Nets
problem

Air Nets

Age
7 to 18
Challenge level
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Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Ding Dong Bell
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.
Placeholder: several colourful numbers
problem

Triangles in the middle

Age
11 to 18
Challenge level
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This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
The Bridges of Konigsberg
problem

The Bridges of Konigsberg

Age
11 to 18
Challenge level
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Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.
Yih or Luk tsut k'i or Three Men's Morris
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.
Instant Insanity
problem

Instant Insanity

Age
11 to 18
Challenge level
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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.

Curve Hunter
problem

Curve Hunter

Age
14 to 18
Challenge level
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This problem challenges you to sketch curves with different properties.

Vector journeys
problem
Favourite

Vector journeys

Age
14 to 18
Challenge level
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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?
Summing geometric progressions
problem
Favourite

Summing geometric progressions

Age
14 to 18
Challenge level
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Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

Can you traverse it?
problem

Can you traverse it?

Age
14 to 18
Challenge level
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How can you decide if a graph is traversable?

Vector walk
problem
Favourite

Vector walk

Age
14 to 18
Challenge level
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Starting with two basic vector steps, which destinations can you reach on a vector walk?
Placeholder: several colourful numbers
problem

Bent out of Shape

Age
14 to 18
Challenge level
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An introduction to bond angle geometry.
What's that graph?
problem
Favourite

What's that graph?

Age
14 to 18
Challenge level
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Can you work out which processes are represented by the graphs?

Iff
problem
Favourite

Iff

Age
14 to 18
Challenge level
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Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Tetra Square
problem

Tetra Square

Age
14 to 18
Challenge level
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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.
Vanishing point
problem

Vanishing point

Age
14 to 18
Challenge level
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How can visual patterns be used to prove sums of series?
Always Perfect
problem
Favourite

Always Perfect

Age
14 to 18
Challenge level
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Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
3D Treasure Hunt
problem

3D Treasure Hunt

Age
14 to 18
Challenge level
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Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?
Back fitter
problem
Favourite

Back fitter

Age
14 to 18
Challenge level
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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?

A Rolling Disc - Periodic Motion
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?
Painting by Numbers
problem

Painting by Numbers

Age
16 to 18
Challenge level
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How many different colours of paint would be needed to paint these pictures by numbers?
Painting by functions
problem

Painting by functions

Age
16 to 18
Challenge level
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Use functions to create minimalist versions of works of art.
Set Square
problem

Set Square

Age
16 to 18
Challenge level
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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?
Wrapping Gifts
problem

Wrapping Gifts

Age
16 to 18
Challenge level
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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?
Stonehenge
problem

Stonehenge

Age
16 to 18
Challenge level
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Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Escriptions
problem

Escriptions

Age
16 to 18
Challenge level
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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
Middle Man
problem

Middle Man

Age
16 to 18
Challenge level
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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?
Polar Bearings
problem

Polar Bearings

Age
16 to 18
Challenge level
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What on earth are polar coordinates, and why would you want to use them?
Parabella
problem
Favourite

Parabella

Age
16 to 18
Challenge level
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This is a beautiful result involving a parabola and parallels.

Trig reps
problem

Trig reps

Age
16 to 18
Challenge level
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Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?
Classic cube
problem

Classic cube

Age
16 to 18
Challenge level
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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?
Fitting flat shapes
problem

Fitting flat shapes

Age
16 to 18
Challenge level
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How efficiently can various flat shapes be fitted together?
Coordinated crystals
problem

Coordinated crystals

Age
16 to 18
Challenge level
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Explore the lattice and vector structure of this crystal.
Classical Means
problem

Classical Means

Age
16 to 18
Challenge level
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Use the diagram to investigate the classical Pythagorean means.
Mach Attack
problem

Mach Attack

Age
16 to 18
Challenge level
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Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.
Maths Shop Window
problem

Maths Shop Window

Age
16 to 18
Challenge level
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Make a functional window display which will both satisfy the manager and make sense to the shoppers
Circles ad infinitum
problem
Favourite

Circles ad infinitum

Age
16 to 18
Challenge level
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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?
Areas and Ratios
problem
Favourite

Areas and Ratios

Age
16 to 18
Challenge level
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Do you have enough information to work out the area of the shaded quadrilateral?
Five circuits, seven spins
problem

Five circuits, seven spins

Age
16 to 18
Challenge level
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A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.
Hyperbolic thinking
problem

Hyperbolic thinking

Age
16 to 18
Challenge level
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Explore the properties of these two fascinating functions using trigonometry as a guide.
Curvy Equation
problem

Curvy Equation

Age
16 to 18
Challenge level
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This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.

Euler meets Schlegel
problem

Euler meets Schlegel

Age
16 to 18
Challenge level
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Discover how networks can be used to prove Euler's Polyhedron formula.

Maximum Scattering
problem

Maximum Scattering

Age
16 to 18
Challenge level
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Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
Rational request
problem

Rational request

Age
16 to 18
Challenge level
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Can you make a curve to match my friend's requirements?

Sheep in wolf's clothing
problem

Sheep in wolf's clothing

Age
16 to 18
Challenge level
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Can you work out what simple structures have been dressed up in these advanced mathematical representations?
Cheese cutting
problem

Cheese cutting

Age
16 to 18
Challenge level
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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?
Ford Circles
problem

Ford Circles

Age
16 to 18
Challenge level
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Can you find the link between these beautiful circle patterns and Farey Sequences?