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 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?

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?

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?

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?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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 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?

Can you use this information to work out Charlie's house number?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

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?

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?

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

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

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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

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

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?

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

Investigate the different ways you could split up these rooms so that you have double the number.

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

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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

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

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?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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

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.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

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?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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

This task follows on from Build it Up and takes the ideas into three dimensions!

This dice train has been made using specific rules. How many different trains can you make?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.