Visualising - Upper Secondary
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?
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
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
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'.
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?
Baravelle
What can you see? What do you notice? What questions can you ask?
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?
Cubic Conundrum
Which of the following cubes can be made from these nets?
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?
Air Nets
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Sprouts
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Diamond Collector
Collect as many diamonds as you can by drawing three straight lines.
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?
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.
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.
Shaping the Universe II - the Solar System
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
Charting Success
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
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.
Charting More Success
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
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.
Marbles in a Box
How many winning lines can you make in a three-dimensional version of noughts and crosses?
LOGO Challenge - Circles as Animals
See if you can anticipate successive 'generations' of the two animals shown here.
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.
Hamiltonian Cube
Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.
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?
Cuboid Challenge
What's the largest volume of box you can make from a square of paper?
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.
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.
Triangles in the Middle
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
The Bridges of Konigsberg
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
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.
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?
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?
Sliced
An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?
Triangles Within Pentagons
Show that all pentagonal numbers are one third of a triangular number.
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?
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?
Painted Purple
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?
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?
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.
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?
Circuit Training
Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever meet at the start again? If so, after how many circuits?
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 ?
Partly Painted Cube
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?
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?
One and Three
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?
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?
Jam
To avoid losing think of another very well known game where the patterns of play are similar.
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?
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?
Picture Story
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Doesn't Add Up
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
A Tilted Square
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Wari
This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?
Something in Common
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
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?
Summing Squares
Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?
Facial Sums
Can you make the numbers around each face of this solid add up to the same total?
Coke Machine
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...
Proximity
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
Fermat's Poser
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
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?
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?
Packing 3D Shapes
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Centre Square
What does Pythagoras' Theorem tell you about the radius of these circles?
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?
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?
Natural Sum
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.
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?
Around and Back
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns around and heads back to the starting point where he meets the runner who is just finishing his first circuit. Find the ratio of their speeds.
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?
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?
Folding in Half
How does the perimeter change when we fold this isosceles triangle in half?
Building Tetrahedra
Can you make a tetrahedron whose faces all have the same perimeter?
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?
Attractive Tablecloths
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Pick's Theorem
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Out of the Window
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.
Triangles Within Triangles
Can you find a rule which connects consecutive triangular numbers?
Rectangle Rearrangement
A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?
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?
Newspaper Sheets
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?
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?
Platonic Planet
Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?
Star Gazing
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
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.
Inside Out
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you can colour every face of all of the smaller cubes?
Triangles Within Squares
Can you find a rule which relates triangular numbers to square numbers?
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?
Building Gnomons
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Mystic Rose
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Trisected Triangle
Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
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.