Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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. . . .

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

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?

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?

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

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?

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?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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

An activity making various patterns with 2 x 1 rectangular tiles.

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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

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

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.

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?

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....?

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?

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

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

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

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

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

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 make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you find all the different ways of lining up these Cuisenaire rods?

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

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?

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?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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

Use the numbers and symbols to make this number sentence correct. How many different ways can you find?

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

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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?

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

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?

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

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?