Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

Use the information about Sally and her brother to find out how many children there are in the Brown family.

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

Can you each work out the number on your card? What do you notice? How could you sort the cards?

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

Use this grid to shade the numbers in the way described. Which numbers do you have left? Do you know what they are called?

This challenge is a game for two players. Choose two numbers from the grid and multiply or divide, then mark your answer on the number line. Can you get four in a row before your partner?

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

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

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 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?

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

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?

Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

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?

Shut the Box game for an adult and child. Can you turn over the cards which match the numbers on the dice?

Four Go game for an adult and child. Will you be the first to have four numbers in a row on the number line?

What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.

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

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

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

Number problems at primary level that require careful consideration.

Can you complete this jigsaw of the multiplication square?

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!

56 406 is the product of two consecutive numbers. What are these two numbers?

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.

Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

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

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

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

Take the number 6 469 693 230 and divide it by the first ten prime numbers and you'll find the most beautiful, most magic of all numbers. What is it?

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?