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?
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?
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?
Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?
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?
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
You have 5 darts and your target score is 44. How many different ways could you score 44?
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?
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
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.
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 do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
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 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?
Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?
Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?
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 arrange 5 different digits (from 0 - 9) in the cross in the way described?
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!
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
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?
Using the statements, can you work out how many of each type of rabbit there are in these pens?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred 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.
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.
There are lots of different methods to find out what the shapes are worth - how many can you find?
This task follows on from Build it Up and takes the ideas into three dimensions!
There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?
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!
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?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
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?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
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?
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.
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?
There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.
In how many ways can you stack these rods, following the rules?
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.