If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Have a go at balancing this equation. Can you find different ways of doing it?

Can you replace the letters with numbers? Is there only one solution in each case?

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

Can you work out some different ways to balance this equation?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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.

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?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Can you order the digits from 1-6 to make a number which is divisible by 6 so when the last digit is removed it becomes a 5-figure number divisible by 5, and so on?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Using all ten cards from 0 to 9, rearrange them to make five prime numbers. Can you find any other ways of doing it?

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

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?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

Investigate the different ways you could split up these rooms so that you have double the number.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Can you use the information to find out which cards I have used?

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?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

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.

Can you make square numbers by adding two prime numbers together?

This Sudoku, based on differences. Using the one clue number can you find the solution?

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?

A few extra challenges set by some young NRICH members.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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.

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!

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.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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.