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

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Can you find the chosen number from the grid using the clues?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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?

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?

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.

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....?

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

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?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

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!

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

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

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

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.

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!

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

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

Try this matching game which will help you recognise different ways of saying the same time interval.

In this matching game, you have to decide how long different events take.

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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

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

What two-digit numbers can you make with these two dice? What can't you make?

Can you find all the ways to get 15 at the top of this triangle of numbers?

This task follows on from Build it Up and takes the ideas into three dimensions!

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.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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

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

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

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?

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