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. . . .
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. . . .
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. . . .
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.
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
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.
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 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
Square It game for an adult and child. Can you come up with a way of always winning this game?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
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!
Show that all pentagonal numbers are one third of a triangular number.
When dice land edge-up, we usually roll again. But what if we
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
Can you find a rule which connects consecutive triangular numbers?
A game for 2 players
Can you discover whether this is a fair game?
Can you find a rule which relates triangular numbers to square numbers?
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?
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?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
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. . . .
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?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
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 mark 4 points on a flat surface so that there are only two
different distances between them?
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 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. . . .
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
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?
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. . . .
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?
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. . . .
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?
A tilted square is a square with no horizontal sides. Can you
devise a general instruction for the construction of a square when
you are given just one of its sides?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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 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.
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?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
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?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?