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.
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
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.
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!
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. . . .
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.
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
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. . . .
Can you find a rule which relates triangular numbers to square numbers?
Can you discover whether this is a fair game?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Show that all pentagonal numbers are one third of a triangular number.
A game for 2 players
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. . . .
Can you find a rule which connects consecutive triangular numbers?
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?
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
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.
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
How efficiently can you pack together disks?
The net of a cube is to be cut from a sheet of card 100 cm square.
What is the maximum volume cube that can be made from a single
piece of card?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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?
What can you see? What do you notice? What questions can you ask?
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. . . .
Consider a watch face which has identical hands and identical marks
for the hours. It is opposite to a mirror. When is the time as read
direct and in the mirror exactly the same between 6 and 7?
Takes you through the systematic way in which you can begin to
solve a mixed up Cubic Net. How close will you come to a solution?
A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
A cube is made from smaller cubes, 5 by 5 by 5, then some of those
cubes are removed. Can you make the specified shapes, and what is
the most and least number of cubes required ?
A bicycle passes along a path and leaves some tracks. Is it
possible to say which track was made by the front wheel and which
by the back wheel?
Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
A triangle PQR, right angled at P, slides on a horizontal floor
with Q and R in contact with perpendicular walls. What is the locus
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Use the diagram to investigate the classical Pythagorean means.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.