Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

These practical challenges are all about making a 'tray' and covering it with paper.

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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.

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

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

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

This article for primary teachers suggests ways in which to help children become better at working systematically.

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

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

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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?

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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

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.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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

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

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

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.

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

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

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?

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

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

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.

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?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

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

What happens when you round these three-digit numbers to the nearest 100?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?