This train line has two tracks which cross at different points. Can
you find all the routes that end at Cheston?
How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?
Moira is late for school. What is the shortest route she can take from the school gates to the entrance?
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did he do it?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Imagine that the puzzle pieces of a jigsaw are roughly a
rectangular shape and all the same size. How many different puzzle
pieces could there be?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
Use the information to describe these marbles. What colours must be
on marbles that sparkle when rolling but are dark inside?
Can you find out in which order the children are standing in this
My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?
In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
Lorenzie was packing his bag for a school trip. He packed four
shirts and three pairs of pants. "I will be able to have a
different outfit each day", he said. How many days will Lorenzie be
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?
In this maze of hexagons, you start in the centre at 0. The next
hexagon must be a multiple of 2 and the next a multiple of 5. What
are the possible paths you could take?
My coat has three buttons. How many ways can you find to do up all
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Chandra, Jane, Terry and Harry ordered their lunches from the
sandwich shop. Use the information below to find out who ordered
My briefcase has a three-number combination lock, but I have
forgotten the combination. I remember that there's a 3, a 5 and an
8. How many possible combinations are there to try?
In how many ways could Mrs Beeswax put ten coins into her three
puddings so that each pudding ended up with at least two coins?
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
Start with three pairs of socks. Now mix them up so that no
mismatched pair is the same as another mismatched pair. Is there
more than one way to do it?
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
Alice and Brian are snails who live on a wall and can only travel
along the cracks. Alice wants to go to see Brian. How far is the
shortest route along the cracks? Is there more than one way to go?
El Crico the cricket has to cross a square patio to get home. He
can jump the length of one tile, two tiles and three tiles. Can you
find a path that would get El Crico home in three jumps?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99
How many ways can you do it?
Move from the START to the FINISH by moving across or down to the
next square. Can you find a route to make these totals?
Zumf makes spectacles for the residents of the planet Zargon, who
have either 3 eyes or 4 eyes. How many lenses will Zumf need to
make all the different orders for 9 families?
How many different shapes can you make by putting four right-
angled isosceles triangles together?
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 problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Using the cards 2, 4, 6, 8, +, - and =, what number statements can
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
In the planet system of Octa the planets are arranged in the shape
of an octahedron. How many different routes could be taken to get
from Planet A to Planet Zargon?
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
Your challenge is to find the longest way through the network
following this rule. You can start and finish anywhere, and with
any shape, as long as you follow the correct order.
What two-digit numbers can you make with these two dice? What can't you make?
The brown frog and green frog want to swap places without getting
wet. They can hop onto a lily pad next to them, or hop over each
other. How could they do it?
These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.
The Red Express Train usually has five red carriages. How many ways
can you find to add two blue carriages?
Explore the different snakes that can be made using 5 cubes.
You have 5 darts and your target score is 44. How many different
ways could you score 44?
Can you arrange 5 different digits (from 0 - 9) in the cross in the