How many ways can you find of tiling the square patio, using square tiles of different sizes?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

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

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

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

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

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?

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?

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?

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?

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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

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

How many trapeziums, of various sizes, are hidden in this picture?

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

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

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.

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

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

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

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.

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

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.

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.

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.?

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

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?

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

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?

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

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

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?

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.

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

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?

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.