Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
How efficiently can various flat shapes be fitted together?
This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.
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?
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.
How efficiently can you pack together disks?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.
A game for 2 players
For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
To avoid losing think of another very well known game where the patterns of play are similar.
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
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?
Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Can you describe this route to infinity? Where will the arrows take you next?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
See if you can anticipate successive 'generations' of the two animals shown here.
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
Can you find a rule which relates triangular numbers to square numbers?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
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 moves.
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. . . .
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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. . . .
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
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?
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
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?
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
Show that all pentagonal numbers are one third of a triangular number.
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
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 circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.
A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?
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?
Can you find a rule which connects consecutive triangular numbers?
A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?
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 of P?
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?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?