Visualising - Upper Secondary

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'.

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?

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.

problem

Prime Magic

Age

7 to 16

Challenge level

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

Favourite

Fruity Totals

Age

7 to 16

Challenge level

In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?

problem

Cubic Conundrum

Age

7 to 16

Challenge level

Which of the following cubes can be made from these nets?

problem

Like a Circle in a Spiral

Age

7 to 16

Challenge level

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

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

Clocking off

Age

7 to 16

Challenge level

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?

Age

11 to 16

Challenge level

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

game

Sliding Puzzle

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

problem

Who's who?

Age

11 to 16

Challenge level

Can you solve the clues to find out who's who on the friendship graph?

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

Favourite

Marbles in a box

Age

11 to 16

Challenge level

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

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.

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

Favourite

Semi-regular Tessellations

Age

11 to 16

Challenge level

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

problem

11x11 square

Age

11 to 16

Challenge level

Here's a neat trick you can do with an 11 by 11 square...

problem

Favourite

Pythagoras Proofs

Age

11 to 16

Challenge level

Can you make sense of these three proofs of Pythagoras' Theorem?

problem

Favourite

Parallelogram It

Age

11 to 16

Challenge level

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.

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

Favourite

Charlie's delightful machine

Age

11 to 16

Challenge level

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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.

problem

Charting success

Age

11 to 16

Challenge level

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

problem

Favourite

Funny Factorisation

Age

11 to 16

Challenge level

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

problem

Favourite

Rhombus It

Age

11 to 16

Challenge level

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.

game

Sprouts

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

problem

Favourite

What's it worth?

Age

11 to 16

Challenge level

There are lots of different methods to find out what the shapes are worth - how many can you find?

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 more success

Age

11 to 16

Challenge level

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

problem

Favourite

Multiple Surprises

Age

11 to 16

Challenge level

Sequences of multiples keep cropping up...

problem

LOGO Challenge - Circles as animals

Age

11 to 16

Challenge level

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

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.

problem

Favourite

Nine Colours

Age

11 to 16

Challenge level

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

LOGO Challenge - Triangles-Squares-Stars

Age

11 to 16

Challenge level

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.

problem

Favourite

Searching for mean(ing)

Age

11 to 16

Challenge level

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

Favourite

Hamiltonian Cube

Age

11 to 16

Challenge level

Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.

problem

Favourite

Cuboid Challenge

Age

11 to 16

Challenge level

What's the largest volume of box you can make from a square of paper?

game

The Triangle Game

Can you discover whether this is a fair game?

problem

Favourite

Take Three From Five

Age

11 to 16

Challenge level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

problem

Concrete calculation

Age

14 to 16

Challenge level

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

Favourite

Slick Summing

Age

14 to 16

Challenge level

Watch the video to see how Charlie works out the sum. Can you adapt his method?

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

Favourite

Which is cheaper?

Age

14 to 16

Challenge level

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

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

Relative Time

Age

14 to 16

Challenge level

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

problem

Star Gazing

Age

14 to 16

Challenge level

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

problem

Twelve Cubed

Age

14 to 16

Challenge level

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

Travelator

Age

14 to 16

Challenge level

When Andrew arrives at the end of the walkway, how far is he ahead of Bill?

problem

Tennis Training

Age

14 to 16

Challenge level

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

problem

Favourite

Speeding boats

Age

14 to 16

Challenge level

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

Painted Purple

Age

14 to 16

Challenge level

Three faces of a $3 \times 3$ cube are painted red, and the other three are painted blue. How many of the 27 smaller cubes have at least one red and at least one blue face?

problem

Favourite

Pair Products

Age

14 to 16

Challenge level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

problem

Favourite

Which is bigger?

Age

14 to 16

Challenge level

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

problem

Slippage

Age

14 to 16

Challenge level

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

Eulerian

Age

14 to 16

Challenge level

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?

problem

Penta Colour

Age

14 to 16

Challenge level

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

Dicey Directions

Age

14 to 16

Challenge level

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

Tilting Triangles

Age

14 to 16

Challenge level

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

Favourite

Partly Painted Cube

Age

14 to 16

Challenge level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

problem

Favourite

Perfectly Square

Age

14 to 16

Challenge level

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

problem

Favourite

Bendy Quad

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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

Favourite

Spotting the loophole

Age

14 to 16

Challenge level

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

Can you traverse it?

Age

14 to 18

Challenge level

How can you decide if a graph is traversable?

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

Changing Places

Age

14 to 16

Challenge level

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

Bent out of Shape

Age

14 to 18

Challenge level

An introduction to bond angle geometry.

problem

Contact

Age

14 to 16

Challenge level

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

Triangles within Triangles

Age

14 to 16

Challenge level

Can you find a rule which connects consecutive triangular numbers?

problem

In or Out?

Age

14 to 16

Challenge level

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?

problem

Tied up

Age

14 to 16

Challenge level

How much of the field can the animals graze?

problem

Making Tracks

Age

14 to 16

Challenge level

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

Vanishing point

Age

14 to 18

Challenge level

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

problem

Overlap

Age

14 to 16

Challenge level

A red square and a blue square overlap. Is the area of the overlap always the same?

problem

Favourite

Painted Cube

Age

14 to 16

Challenge level

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

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

Picture Story

Age

14 to 16

Challenge level

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

problem

Triangles within Squares

Age

14 to 16

Challenge level

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

problem

Trisected Triangle

Age

14 to 16

Challenge level

Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

problem

Favourite

One and three

Age

14 to 16

Challenge level

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400 metres from B. How long is the lake?

problem

One Out One Under

Age

14 to 16

Challenge level

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?

article

Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

problem

Tetrahedra Tester

Age

14 to 16

Challenge level

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

Favourite

Factorising with Multilink

Age

14 to 16

Challenge level

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

problem

Proximity

Age

14 to 16

Challenge level

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

problem

Triangles within Pentagons

Age

14 to 16

Challenge level

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

problem

Favourite

Quadratic Patterns

Age

14 to 16

Challenge level

Surprising numerical patterns can be explained using algebra and diagrams...

problem

Favourite

Plus Minus

Age

14 to 16

Challenge level

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

problem

The perforated cube

Age

14 to 16

Challenge level

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

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

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

Favourite

What's that graph?

Age

14 to 18

Challenge level

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

problem

Natural Sum

Age

14 to 16

Challenge level

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.