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?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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?

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?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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

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

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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?

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. . . .

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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'.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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

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

How many models can you find which obey these rules?

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

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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

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

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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

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

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?

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

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

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.

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?

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

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

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?

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

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.