On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?
Every day at noon a boat leaves Le Havre for New York while another
boat leaves New York for Le Havre. The ocean crossing takes seven
days. How many boats will each boat cross during their journey?
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. . . .
A bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
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?
At the time of writing the hour and minute hands of my clock are at
right angles. How long will it be before they are at right angles
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
Anne completes a circuit around a circular track in 40 seconds.
Brenda runs in the opposite direction and meets Anne every 15
seconds. How long does it take Brenda to run around the track?
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. . . .
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
Which hexagons tessellate?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
In this problem, we have created a pattern from smaller and smaller
squares. If we carried on the pattern forever, what proportion of
the image would be coloured blue?
How efficiently can you pack together disks?
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?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
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.
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
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. . . .
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
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.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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!
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
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.
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
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
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 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.
Can you find a rule which relates triangular numbers to square numbers?
Show that all pentagonal numbers are one third of a triangular number.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you find a rule which connects consecutive triangular numbers?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
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?
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?
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. . . .
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. . . .