Can you discover whether this is a fair game?
To avoid losing think of another very well known game where the
patterns of play are similar.
A game for 2 players
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
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
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. . . .
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
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. . . .
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Can you find a rule which connects consecutive triangular numbers?
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which relates triangular numbers to square numbers?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
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?
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?
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. . . .
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.
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
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. . . .
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
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!
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
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. . . .
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.
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?
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
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.
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. . . .
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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?
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.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
A huge wheel is rolling past your window. What do you see?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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 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?
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. . . .
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?