Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
A game for 2 players
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 students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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.
Can you describe this route to infinity? Where will the arrows take you next?
Can you make a tetrahedron whose faces all have the same perimeter?
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
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. . . .
Can you find a rule which connects consecutive triangular numbers?
Square It game for an adult and child. Can you come up with a way of always winning this game?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which relates triangular numbers to square numbers?
Can you discover whether this is a fair game?
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 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. . . .
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.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
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?
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
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.
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.
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. . . .
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
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. . . .
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 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?
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. . . .
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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.
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?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
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?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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?
This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .
What can you see? What do you notice? What questions can you ask?
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?
Small circles nestle under touching parent circles when they sit on
the axis at neighbouring points in a Farey sequence.
Use the diagram to investigate the classical Pythagorean means.
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?