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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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?
Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?
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. . . .
A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
How efficiently can you pack together disks?
Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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.
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?
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?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
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. . . .
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
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 stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?
A game for 2 players
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?
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 task depends on groups working collaboratively, discussing and reasoning to agree a final product.
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?
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. . . .
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
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.
Can you find a rule which relates triangular numbers to square numbers?
Can you find a rule which connects consecutive triangular numbers?
Show that all pentagonal numbers are one third of a triangular number.
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?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
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?
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?
What can you see? What do you notice? What questions can you ask?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
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.
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?
See if you can anticipate successive 'generations' of the two animals shown here.
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?
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 of P?
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.
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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. . . .