This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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....?

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

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

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?

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

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

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

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

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

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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

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?

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?

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

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

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?

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

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?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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?

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

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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?

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. . . .

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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.

Given the products of adjacent cells, can you complete this Sudoku?

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?

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!

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.

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?

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

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

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

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

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

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