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 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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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.
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?
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 seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.
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?
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?
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
Investigate the different ways you could split up these rooms so that you have double the number.
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?
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?
This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.
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?
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.
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.
Have a go at balancing this equation. Can you find different ways of doing it?
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.
Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?
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?
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.
These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.