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?

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

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?

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

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?

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

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

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?

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

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?

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

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

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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?

Use the numbers and symbols to make this number sentence correct. How many different ways can you find?

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?

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

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

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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?

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

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

What is the best way to shunt these carriages so that each train can continue its journey?

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.

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

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

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

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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.

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.

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.

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?

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?

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?

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

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

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?

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.

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 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.

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

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?