What happens when you round these three-digit numbers to the nearest 100?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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?

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

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

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

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?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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 if you work in a systematic way, you won't leave any out.

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

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

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

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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

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?

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.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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?

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

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

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

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?

Find out about Magic Squares in this article written for students. Why are they magic?!

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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?

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Find out what a "fault-free" rectangle is and try to make some of your own.

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

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?

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?

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

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?