A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
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.
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
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?
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?
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after
every cut you can rearrange the pieces before cutting straight
through, can you do it in fewer?
A useful visualising exercise which offers opportunities for
discussion and generalising, and which could be used for thinking
about the formulae needed for generating the results on a
Find all the ways to cut out a 'net' of six squares that can be
folded into a cube.
A package contains a set of resources designed to develop pupils'
mathematical thinking. This package places a particular emphasis on
“visualising” and is designed to meet the needs. . . .
A triangle ABC resting on a horizontal line is "rolled" along the
line. Describe the paths of each of the vertices and the
relationships between them and the original triangle.
Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?
How can you make an angle of 60 degrees by folding a sheet of paper
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
When dice land edge-up, we usually roll again. But what if we
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
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?
Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface. . . .
What is the shape of wrapping paper that you would need to completely wrap this model?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
Can you find a way of representing these arrangements of balls?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
A huge wheel is rolling past your window. What do you see?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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.
Draw all the possible distinct triangles on a 4 x 4 dotty grid.
Convince me that you have all possible triangles.
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
Can you work out what kind of rotation produced this pattern of
pegs in our pegboard?
What is the greatest number of squares you can make by overlapping
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?
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
A rectangular field has two posts with a ring on top of each post.
There are two quarrelsome goats and plenty of ropes which you can
tie to their collars. How can you secure them so they can't. . . .
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
This article for teachers discusses examples of problems in which
there is no obvious method but in which children can be encouraged
to think deeply about the context and extend their ability to. . . .
Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.
Can you explain why it is impossible to construct this triangle?
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.
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
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
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
Have a go at this 3D extension to the Pebbles problem.
Can you picture where this letter "F" will be on the grid if you
flip it in these different ways?