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. . . .
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?
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?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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?
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?
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.
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?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
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.
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?
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
How efficiently can you pack together disks?
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
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. . . .
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 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. . . .
To avoid losing think of another very well known game where the
patterns of play are similar.
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
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. . . .
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?
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
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
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?
What is the shape of wrapping paper that you would need to completely wrap this model?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Show that all pentagonal numbers are one third of a triangular number.
A game for 2 players
Can you describe this route to infinity? Where will the arrows take you next?
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
Can you find a rule which connects consecutive triangular numbers?
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.
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?
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. . . .
Take a line segment of length 1. Remove the middle third. Remove
the middle thirds of what you have left. Repeat infinitely many
times, and you have the Cantor Set. Can you picture it?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
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!
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?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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.
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
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?
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?