If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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.

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

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

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

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

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

Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

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

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

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

Find all the numbers that can be made by adding the dots on two dice.

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?

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

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?

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?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

What happens when you try and fit the triomino pieces into these two grids?

Can you find out in which order the children are standing in this line?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

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

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. . . .

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

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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?

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

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?

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

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?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

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.

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

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?

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

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

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

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

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?

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

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

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

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