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

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?

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

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

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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?

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

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?

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.?

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

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?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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

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

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

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?

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

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?

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

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

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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?

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

How many different triangles can you make on a circular pegboard that has nine pegs?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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.

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

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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

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

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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?

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

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?

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?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

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