In this article, the NRICH team describe the process of selecting solutions for publication on the site.
This article for primary teachers suggests ways in which to help children become better at working systematically.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
How could you arrange at least two dice in a stack so that the total of the visible spots is 18?
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?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
This challenge extends the Plants investigation so now four or more children are involved.
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
In this town, houses are built with one room for each person. There
are some families of seven people living in the town. In how many
different ways can they build their houses?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
A package contains a set of resources designed to develop
students’ mathematical thinking. This package places a
particular emphasis on “being systematic” and is
designed to meet. . . .
George and Jim want to buy a chocolate bar. George needs 2p more
and Jim need 50p more to buy it. How much is the chocolate bar?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Look carefully at the numbers. What do you notice? Can you make
another square using the numbers 1 to 16, that displays the same
Use your logical-thinking skills to deduce how much Dan's crisps
and ice-cream cost altogether.
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.
Systematically explore the range of symmetric designs that can be
created by shading parts of the motif below. Use normal square
lattice paper to record your results.
How many triangles can you make on the 3 by 3 pegboard?
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?
I was in my car when I noticed a line of four cars on the lane next
to me with number plates starting and ending with J, K, L and M.
What order were they in?
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?
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?
Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.
Alice's mum needs to go to each child's house just once and then
back home again. How many different routes are there? Use the
information to find out how long each road is on the route she
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
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?
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Tim had nine cards each with a different number from 1 to 9 on it.
How could he have put them into three piles so that the total in
each pile was 15?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
The Vikings communicated in writing by making simple scratches on
wood or stones called runes. Can you work out how their code works
using the table of the alphabet?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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....?
Use these head, body and leg pieces to make Robot Monsters which
are different heights.
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?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
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?
Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Can you cover the camel with these pieces?
How many different ways can you find to join three equilateral
triangles together? Can you convince us that you have found them
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?
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
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.