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?
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
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?
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?
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
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?
What can you see? What do you notice? What questions can you ask?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
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?
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?
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?
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
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. . . .
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?
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 ?
What's the largest volume of box you can make from a square of paper?
Can you make a tetrahedron whose faces all have the same perimeter?
Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?
How much of the field can the animals graze?
Can you work out the dimensions of the three cubes?
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?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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.
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?
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.
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
To avoid losing think of another very well known game where the patterns of play are similar.
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 square.
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.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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.
Which of the following cubes can be made from these nets?
Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?
How can visual patterns be used to prove sums of series?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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?
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. . . .
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
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. . . .
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
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?
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
A game for 2 players
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?
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?
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?