Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Can you find a rule which connects consecutive triangular numbers?
Show that all pentagonal numbers are one third of a triangular number.
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
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you find a rule which relates triangular numbers to square numbers?
A game for 2 players
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
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?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
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 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
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
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. . . .
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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.
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 reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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?
Have a go at this 3D extension to the Pebbles problem.
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?
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?
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
A huge wheel is rolling past your window. What do you see?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
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. . . .
To avoid losing think of another very well known game where the
patterns of play are similar.
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
A spider is sitting in the middle of one of the smallest walls in a
room and a fly is resting beside the window. What is the shortest
distance the spider would have to crawl to catch the fly?
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
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?
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?
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?
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. . . .
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.
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.