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?
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?
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
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?
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?
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
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Using different numbers of sticks, how many different triangles are
you able to make? Can you make any rules about the numbers of
sticks that make the most triangles?
Is it possible to place 2 counters on the 3 by 3 grid so that there
is an even number of counters in every row and every column? How
about if you have 3 counters or 4 counters or....?
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
Can you find all the different ways of lining up these Cuisenaire
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
My cousin was 24 years old on Friday April 5th in 1974. On what day
of the week was she born?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
In this matching game, you have to decide how long different events take.
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
These practical challenges are all about making a 'tray' and covering it with paper.
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
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?
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 problem is based on a code using two different prime numbers
less than 10. You'll need to multiply them together and shift the
alphabet forwards by the result. Can you decipher the code?
A few extra challenges set by some young NRICH members.
An extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Can you find all the different triangles on these peg boards, and
find their angles?
The challenge here is to find as many routes as you can for a fence
to go so that this town is divided up into two halves, each with 8
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Mr McGregor has a magic potting shed. Overnight, the number of
plants in it doubles. He'd like to put the same number of plants in
each of three gardens, planting one garden each day. Can he do it?