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?
Are these statements always true, sometimes true or never true?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Got It game for an adult and child. How can you play so that you know you will always win?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
This activity involves rounding four-digit numbers to the nearest thousand.
This challenge asks you to imagine a snake coiling on itself.
This task follows on from Build it Up and takes the ideas into three dimensions!
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
Here are two kinds of spirals for you to explore. What do you notice?
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?
An investigation that gives you the opportunity to make and justify
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?
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
Can you find all the ways to get 15 at the top of this triangle of numbers?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Find the sum of all three-digit numbers each of whose digits is
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
Can you explain the strategy for winning this game with any target?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...