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?
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?
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?
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?
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?
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
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
What is the best way to shunt these carriages so that each train
can continue its journey?
Can you shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
A toy has a regular tetrahedron, a cube and a base with triangular
and square hollows. If you fit a shape into the correct hollow a
bell rings. How many times does the bell ring in a complete game?
A magician took a suit of thirteen cards and held them in his hand
face down. Every card he revealed had the same value as the one he
had just finished spelling. How did this work?
A tetromino is made up of four squares joined edge to edge. Can
this tetromino, together with 15 copies of itself, be used to cover
an eight by eight chessboard?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
10 space travellers are waiting to board their spaceships. There
are two rows of seats in the waiting room. Using the rules, where
are they all sitting? Can you find all the possible ways?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Can you make a 3x3 cube with these shapes made from small cubes?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
How many different triangles can you make on a circular pegboard that has nine pegs?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
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.
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
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 how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
We start with one yellow cube and build around it to make a 3x3x3
cube with red cubes. Then we build around that red cube with blue
cubes and so on. How many cubes of each colour have we used?
This article for teachers describes how modelling number properties
involving multiplication using an array of objects not only allows
children to represent their thinking with concrete materials,. . . .
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?
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. . . .
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
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?
One face of a regular tetrahedron is painted blue and each of the
remaining faces are painted using one of the colours red, green or
yellow. How many different possibilities are there?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
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.
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
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?
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?