How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

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

Can you work out some different ways to balance this equation?

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

Have a go at balancing this equation. Can you find different ways of doing it?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

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.

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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

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

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?

Arrange 3 red, 3 blue and 3 yellow counters into a three-by-three square grid, so that there is only one of each colour in every row and every column

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.

Can you make square numbers by adding two prime numbers together?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?

The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.

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?

An investigation that gives you the opportunity to make and justify predictions.

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?

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.

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?

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?

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?

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.

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!

Can you draw a square in which the perimeter is numerically equal to the area?

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

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

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

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?

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

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?

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?

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

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?

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

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?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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!

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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.

The pages of my calendar have got mixed up. Can you sort them out?

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?