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?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
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?
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
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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?
What is the best way to shunt these carriages so that each train
can continue its journey?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared 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.
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
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....?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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.
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?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Find the values of the nine letters in the sum: FOOT + BALL = GAME
These practical challenges are all about making a 'tray' and covering it with paper.
Can you find all the different ways of lining up these Cuisenaire
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?
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?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
How many different triangles can you make on a circular pegboard that has nine pegs?
The letters in the following addition sum represent the digits 1
... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
When newspaper pages get separated at home we have to try to sort
them out and get things in the correct order. How many ways can we
arrange these pages so that the numbering may be different?
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
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?
Put 10 counters in a row. Find a way to arrange the counters into
five pairs, evenly spaced in a row, in just 5 moves, using the
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
Try out the lottery that is played in a far-away land. What is the
chance of winning?
Let's say you can only use two different lengths - 2 units and 4
units. Using just these 2 lengths as the edges how many different
cuboids can you make?
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
A few extra challenges set by some young NRICH members.
Investigate the different ways you could split up these rooms so
that you have double the number.
Find out what a "fault-free" rectangle is and try to make some of
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.