At the time of writing the hour and minute hands of my clock are at
right angles. How long will it be before they are at right angles
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
Anne completes a circuit around a circular track in 40 seconds.
Brenda runs in the opposite direction and meets Anne every 15
seconds. How long does it take Brenda to run around the track?
A bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
Every day at noon a boat leaves Le Havre for New York while another
boat leaves New York for Le Havre. The ocean crossing takes seven
days. How many boats will each boat cross during their journey?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Can you maximise the area available to a grazing goat?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
A train leaves on time. After it has gone 8 miles (at 33mph) the
driver looks at his watch and sees that the hour hand is exactly
over the minute hand. When did the train leave the station?
Can you find ways of joining cubes together so that 28 faces are
Can you find a way of representing these arrangements of balls?
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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?
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
Exploring and predicting folding, cutting and punching holes and
Can you cut a regular hexagon into two pieces to make a
parallelogram? Try cutting it into three pieces to make a rhombus!
Can you arrange the shapes in a chain so that each one shares a
face (or faces) that are the same shape as the one that follows it?
Each of the nets of nine solid shapes has been cut into two pieces.
Can you see which pieces go together?
Reasoning about the number of matches needed to build squares that
share their sides.
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
An extension of noughts and crosses in which the grid is enlarged
and the length of the winning line can to altered to 3, 4 or 5.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start.
How many Hamiltonian circuits can you find in these graphs?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Which of these dice are right-handed and which are left-handed?
Can you fit the tangram pieces into the outlines of the watering can and man in a boat?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
What can you see? What do you notice? What questions can you ask?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
Paint a stripe on a cardboard roll. Can you predict what will
happen when it is rolled across a sheet of paper?
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?