This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

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?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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?

Can you substitute numbers for the letters in these sums?

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?

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

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

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

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

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.

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?

What could the half time scores have been in these Olympic hockey matches?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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.

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

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

The Zargoes use almost the same alphabet as English. What does this birthday message say?

In how many ways can you stack these rods, following the rules?

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?

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?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

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

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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?

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?

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.

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?

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

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?

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