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. . . .
Some diagrammatic 'proofs' of algebraic identities and
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Can you use the diagram to prove the AM-GM inequality?
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?
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. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Can you discover whether this is a fair game?
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
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?
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
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
Can you find a rule which relates triangular numbers to square numbers?
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.
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
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?
Use the diagram to investigate the classical Pythagorean means.
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which connects consecutive triangular numbers?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
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!
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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?
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.
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?
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. . . .
Can you make a tetrahedron whose faces all have the same perimeter?
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
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
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?
What can you see? What do you notice? What questions can you ask?
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
How efficiently can you pack together disks?
See if you can anticipate successive 'generations' of the two
animals shown here.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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?
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. . . .