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?
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?
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
Design an arrangement of display boards in the school hall which fits the requirements of different people.
What is the best way to shunt these carriages so that each train
can continue its journey?
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 smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
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 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?
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. . . .
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
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?
How many models can you find which obey these rules?
An activity making various patterns with 2 x 1 rectangular tiles.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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?
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.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
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
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?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
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?
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
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?
Two sudokus in one. Challenge yourself to make the necessary
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
A Sudoku that uses transformations as supporting clues.
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
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.
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 can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Investigate the different ways you could split up these rooms so
that you have double the number.