Find out what a "fault-free" rectangle is and try to make some of
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
Find the sum of all three-digit numbers each of whose digits is
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
An investigation that gives you the opportunity to make and justify
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
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?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
What happens when you round these three-digit numbers to the nearest 100?
Got It game for an adult and child. How can you play so that you know you will always win?
What happens when you round these numbers to the nearest whole number?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
How many centimetres of rope will I need to make another mat just
like the one I have here?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?