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?
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!
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
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?
Takes you through the systematic way in which you can begin to
solve a mixed up Cubic Net. How close will you come to a solution?
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 ?
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?
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
Can you work out the dimensions of the three cubes?
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.
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?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
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?
To avoid losing think of another very well known game where the
patterns of play are similar.
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?
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?
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?
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 game for 2 people. Take turns joining two dots, until your opponent is unable to move.
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
What can you see? What do you notice? What questions can you ask?
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. . . .
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
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.
Show that all pentagonal numbers are one third of a triangular number.
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Which of the following cubes can be made from these nets?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
Can you find a rule which connects consecutive triangular numbers?
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?
A game for 2 players
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.
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?
Can you find a rule which relates triangular numbers to square numbers?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Two intersecting circles have a common chord AB. The point C moves
on the circumference of the circle C1. The straight lines CA and CB
meet the circle C2 at E and F respectively. As the point C. . . .
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?
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. . . .
Can you discover whether this is a fair game?
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?
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?
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
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
In how many different ways can I colour the five edges of a
pentagon red, blue and green so that no two adjacent edges are the
How efficiently can you pack together disks?
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.
Use the diagram to investigate the classical Pythagorean means.