If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?
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 different rectangles can you make using this set of rods?
Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?
Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
In this game you are challenged to gain more columns of lily pads than your opponent.
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
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....?
Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
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.
There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.
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?
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
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?
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.
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?
This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
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?
What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
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!
A few extra challenges set by some young NRICH members.
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?
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 jugs.
On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.