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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
This challenge asks you to imagine a snake coiling on itself.
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.
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?
This activity involves rounding four-digit numbers to the nearest thousand.
Ben’s class were making cutting up number tracks. First they
cut them into twos and added up the numbers on each piece. What
patterns could they see?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten
numbers from the bags above so that their total is 37.
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.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Can you explain how this card trick works?
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Delight your friends with this cunning trick! Can you explain how
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
An investigation that gives you the opportunity to make and justify
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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.
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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Got It game for an adult and child. How can you play so that you know you will always win?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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. . . .
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?